title: Samuel Johnson

text: Two hundred years after Johnson's death, the posthumous diagnosis of Tourette syndrome became widely accepted. The condition was unknown during Johnson's lifetime, but Boswell describes Johnson displaying signs of Tourette syndrome, including tics and other involuntary movements. According to Boswell "he commonly held his head to one side ... moving his body backwards and forwards, and rubbing his left knee in the same direction, with the palm of his hand ... [H]e made various sounds" like "a half whistle" or "as if clucking like a hen", and "... all this accompanied sometimes with a thoughtful look, but more frequently with a smile. Generally when he had concluded a period, in the course of a dispute, by which time he was a good deal exhausted by violence and vociferation, he used to blow out his breath like a whale." There are many similar accounts; in particular, Johnson was said to "perform his gesticulations" at the threshold of a house or in doorways. When asked by a little girl why he made such noises and acted in that way, Johnson responded: "From bad habit." The diagnosis of the syndrome was first made in a 1967 report, and Tourette syndrome researcher Arthur K. Shapiro described Johnson as "the most notable example of a successful adaptation to life despite the liability of Tourette syndrome". Details provided by the writings of Boswell, Hester Thrale, and others reinforce the diagnosis, with one paper concluding:


title: Samuel Johnson

text: Johnson was, in the words of Steven Lynn, "more than a well-known writer and scholar"; he was a celebrity, for the activities and the state of his health in his later years were constantly reported in various journals and newspapers, and when there was nothing to report, something was invented. According to Bate, "Johnson loved biography," and he "changed the whole course of biography for the modern world. One by-product was the most famous single work of biographical art in the whole of literature, Boswell's "Life of Johnson", and there were many other memoirs and biographies of a similar kind written on Johnson after his death." These accounts of his life include Thomas Tyers's "A Biographical Sketch of Dr Samuel Johnson" (1784); Boswell's "The Journal of a Tour to the Hebrides" (1785); Hester Thrale's "Anecdotes of the Late Samuel Johnson", which drew on entries from her diary and other notes; John Hawkins's "Life of Samuel Johnson", the first full-length biography of Johnson; and, in 1792, Arthur Murphy's "An Essay on the Life and Genius of Samuel Johnson", which replaced Hawkins's biography as the introduction to a collection of Johnson's "Works". Another important source was Fanny Burney, who described Johnson as "the acknowledged Head of Literature in this kingdom" and kept a diary containing details missing from other biographies. Above all, Boswell's portrayal of Johnson is the work best known to general readers. Although critics like Donald Greene argue about its status as a true biography, the work became successful as Boswell and his friends promoted it at the expense of the many other works on Johnson's life.


title: Samuel Johnson

text: In criticism, Johnson had a lasting influence, although not everyone viewed him favourably. Some, like Macaulay, regarded Johnson as an idiot savant who produced some respectable works, and others, like the Romantic poets, were completely opposed to Johnson's views on poetry and literature, especially with regard to Milton. However, some of their contemporaries disagreed: Stendhal's "Racine et Shakespeare" is based in part on Johnson's views of Shakespeare, and Johnson influenced Jane Austen's writing style and philosophy. Later, Johnson's works came into favour, and Matthew Arnold, in his "Six Chief Lives from Johnson's "Lives of the Poets"", considered the "Lives" of Milton, Dryden, Pope, Addison, Swift, and Gray as "points which stand as so many natural centres, and by returning to which we can always find our way again".


title: Samuel Johnson

text: More than a century after his death, literary critics such as G. Birkbeck Hill and T. S. Eliot came to regard Johnson as a serious critic. They began to study Johnson's works with an increasing focus on the critical analysis found in his edition of Shakespeare and "Lives of the Poets". Yvor Winters claimed that "A great critic is the rarest of all literary geniuses; perhaps the only critic in English who deserves that epithet is Samuel Johnson". F. R. Leavis agreed and, on Johnson's criticism, said, "When we read him we know, beyond question, that we have here a powerful and distinguished mind operating at first hand upon literature. This, we can say with emphatic conviction, really "is" criticism". Edmund Wilson claimed that "The "Lives of the Poets" and the prefaces and commentary on Shakespeare are among the most brilliant and the most acute documents in the whole range of English criticism".


title: Samuel Johnson

text: The critic Harold Bloom placed Johnson's work firmly within the Western canon, describing him as "unmatched by any critic in any nation before or after him ... Bate in the finest insight on Johnson I know, emphasised that no other writer is so obsessed by the realisation that the mind is an "activity", one that will turn to destructiveness of the self or of others unless it is directed to labour." Johnson's philosophical insistence that the language within literature must be examined became a prevailing mode of literary theory during the mid-20th century.


title: Samuel Johnson

text: Half of Johnson's surviving correspondence, together with some of his manuscripts, editions of his books, paintings and other items associated with him are in the Donald and Mary Hyde Collection of Dr. Samuel Johnson, housed at Houghton Library at Harvard University since 2003. The collection includes drafts of his "Plan for a Dictionary", documents associated with Hester Thrale Piozzi and James Boswell (including corrected proofs of his "Life of Johnson") and a teapot owned by Johnson.


title: Samuel Johnson

text: There are many societies formed around and dedicated to the study and enjoyment of Samuel Johnson's life and works. On the bicentennial of Johnson's death in 1984, Oxford University held a week-long conference featuring 50 papers, and the Arts Council of Great Britain held an exhibit of "Johnsonian portraits and other memorabilia". The London "Times" and "Punch" produced parodies of Johnson's style for the occasion. In 1999, the BBC Four television channel started the Samuel Johnson Prize, an award for non-fiction. A Royal Society of Arts blue plaque, unveiled in 1876, marks Johnson's Gough Square house. In 2009, Johnson was among the ten people selected by the Royal Mail for their "Eminent Britons" commemorative postage stamp issue. On 18 September 2017 Google commemorated Johnson's 308th birthday with a Google Doodle. The date of his death, 13 December, is commemorated in the Church of England's Calendar of Saints. There is a memorial to him at St Paul's Cathedral in London.


title: Tapir

text: Tapirs ( ) are large, herbivorous mammals belonging to the family Tapiridae. They are similar in shape to a pig, with a short, prehensile nose trunk. Tapirs inhabit jungle and forest regions of South and Central America, with one species inhabiting Southeast Asia. They are one of three extant branches of Perissodactyla (odd-toed ungulates), alongside equines and rhinoceros. Only a single genus, "Tapirus" is currently extant. Tapirs migrated into South America during the Pleistocene epoch from North America after the formation of the Isthmus of Panama as part of the Great American Interchange. Tapirs were once widespread in North America until the arrival of humans at the end of the Late Pleistocene, around 12,000 years ago.


title: Tapir

text: There are four widely recognized extant species of tapir, all in the genus Tapirus of the family Tapiridae. They are the South American tapir, the Malayan tapir, Baird's tapir, and the mountain tapir. In 2013, a group of researchers said they had identified a fifth species of tapir, the kabomani tapir. However, the existence of the kabomani tapir as a distinct species has been widely disputed, and recent genetic evidence further suggests that it actually is part of the species South American tapir."


title: Tapir

text: The four species are all classified on the IUCN Red List as Endangered or Vulnerable. The tapirs have a number of extinct relatives in the superfamily Tapiroidea. The closest extant relatives of the tapirs are the other odd-toed ungulates, which include horses, donkeys, zebras and rhinoceroses.


title: Tapir

text: Size varies between types, but most tapirs are about long, stand about 1 m (3 ft) high at the shoulder, and weigh between 150 and 300 kg (330 and 700 lb). Their coats are short and range in colour from reddish brown, to grey, to nearly black, with the notable exceptions of the Malayan tapir, which has a white, saddle-shaped marking on its back, and the mountain tapir, which has longer, woolly fur. All tapirs have oval, white-tipped ears, rounded, protruding rumps with stubby tails, and splayed, hooved toes, with four toes on the front feet and three on the hind feet, which help them to walk on muddy and soft ground. Baby tapirs of all types have striped-and-spotted coats for camouflage. Females have a single pair of mammary glands, and males have long penises relative to their body size.


title: Tapir

text: The proboscis of the tapir is a highly flexible organ, able to move in all directions, allowing the animals to grab foliage that would otherwise be out of reach. Tapirs often exhibit the flehmen response, a posture in which they raise their snouts and show their teeth to detect scents. This response is frequently exhibited by bulls sniffing for signs of other males or females in oestrus in the area. The length of the proboscis varies among species; Malayan tapirs have the longest snouts and Brazilian tapirs have the shortest. The evolution of tapir probosces, made up almost entirely of soft tissues rather than bony internal structures, gives the Tapiridae skull a unique form in comparison to other perissodactyls, with a larger sagittal crest, orbits positioned more rostrally, a posteriorly telescoped cranium, and a more elongated and retracted nasoincisive incisure.


title: Tapir

text: Totaling 42 to 44 teeth, this dentition is closer to that of equids, which may differ by one less canine, than their other perissodactyl relatives, rhinoceroses. Their incisors are chisel-shaped, with the third large, conical upper incisor separated by a short gap from the considerably smaller canine. A much longer gap is found between the canines and premolars, the first of which may be absent. Tapirs are lophodonts, and their cheek teeth have distinct lophs (ridges) between protocones, paracones, metacones and hypocones.


title: Tapir

text: Tapirs have brown eyes, often with a bluish cast to them, which has been identified as corneal cloudiness, a condition most commonly found in Malayan tapirs. The exact etiology is unknown, but the cloudiness may be caused by excessive exposure to light or by trauma. However, the tapir's sensitive ears and strong sense of smell help to compensate for deficiencies in vision.


title: Tapir

text: Young tapirs reach sexual maturity between three and five years of age, with females maturing earlier than males. Under good conditions, a healthy female tapir can reproduce every two years; a single young, called a calf, is born after a gestation of about 13 months. The natural lifespan of a tapir is about 25 to 30 years, both in the wild and in zoos. Apart from mothers and their young offspring, tapirs lead almost exclusively solitary lives.


title: Tapir

text: Although they frequently live in dryland forests, tapirs with access to rivers spend a good deal of time in and under water, feeding on soft vegetation, taking refuge from predators, and cooling off during hot periods. Tapirs near a water source will swim, sink to the bottom, and walk along the riverbed to feed, and have been known to submerge themselves to allow small fish to pick parasites off their bulky bodies. Along with freshwater lounging, tapirs often wallow in mud pits, which also help to keep them cool and free of insects.


title: Tapir

text: In the wild, the tapir's diet consists of fruit, berries, and leaves, particularly young, tender vegetation. Tapirs will spend many of their waking hours foraging along well-worn trails, snouts to the ground in search of food. Baird's tapirs have been observed to eat around 40 kg (85 lb) of vegetation in one day.


title: Tapir

text: Tapirs are largely nocturnal and crepuscular, although the smaller mountain tapir of the Andes is generally more active during the day than its congeners. They have monocular vision.


title: Tapir

text: Copulation may occur in or out of water. In captivity, mating pairs will often copulate several times during oestrus. Intromission lasts between 10 and 20 minutes.


title: Tapir

text: Adult tapirs are large enough to have few natural predators, and the thick skin on the backs of their necks helps to protect them from threats such as jaguars, crocodiles, anacondas, and tigers. The creatures are also able to run fairly quickly, considering their size and cumbersome appearance, finding shelter in the thick undergrowth of the forest or in water. Hunting for meat and hides has substantially reduced their numbers and, more recently, habitat loss has resulted in the conservation watch-listing of all four species; the Brazilian tapir is classified as vulnerable, and Baird's tapir, the mountain tapir, and the Malayan tapir are endangered. According to 2022 study published in the "Neotropical Biology and Conservation", the lowland tapir in the Atlantic Forest is at risk of complete extinction as a result of anthropogenic pressures, in particular hunting, deforestation and population isolation.


title: Tapir

text: The first tapiroids, such as "Heptodon", appeared in the early Eocene of North America. They appeared very similar to modern forms, but were about half the size, and lacked the proboscis. The first true tapirs appeared in the Oligocene. By the Miocene, such genera as "Miotapirus" were almost indistinguishable from the extant species. Asian and American tapirs were believed to have diverged around 20 to 30 million years ago; tapirs later migrated from North America to South America around 3 million years ago, as part of the Great American Interchange. For much of their history, tapirs were spread across the Northern Hemisphere, where they became extinct as recently as 10,000 years ago. "T. merriami", "T. veroensis", "T. copei", and "T. californicus" became extinct during the Pleistocene in North America. The giant tapir survived until about 4,000 years ago in China.


title: Tapir

text: Approximate divergence times based on a 2013 analysis of mtDNA sequences are 0.5 Ma for "T. kabomani" and the "T. terrestris"–"T. pinchaque" clade, 5 Ma for "T. bairdii" and the three South American tapirs, and 9 Ma for the branching of "T. indicus". "T. pinchaque" arises from within a paraphyletic complex of "T. terrestris" populations.


title: Tapir

text: The tapir may have evolved from the paleothere "Hyracotherium" (once thought to be a primitive horse).


title: Tapir

text: The Malayan tapir, the species most isolated geographically and genetically, has a significantly smaller number of chromosomes and has been found to share fewer homologies with the three types of American tapirs. A number of conserved autosomes (13 between karyotypes of Baird's tapir and the South American tapir, and 15 between Baird's and the mountain tapir) have also been found in the American species that are not found in the Asian animal. However, geographic proximity is not an absolute predictor of genetic similarity; for instance, G-banded preparations have revealed Malayan, Baird's and South American tapirs have identical X chromosomes, while mountain tapirs are separated by a heterochromatic addition/deletion.


title: Tapir

text: Lack of genetic diversity in tapir populations has become a major source of concern for conservationists. Habitat loss has isolated already small populations of wild tapirs, putting each group in greater danger of dying out completely. Even in zoos, genetic diversity is limited; all captive mountain tapirs, for example, are descended from only two founder individuals.


title: Tapir

text: Hybrids of Baird's and the South American tapirs were bred at the San Francisco Zoo around 1969 and later produced a backcross second generation.


title: Tapir

text: A number of conservation projects have been started around the world. The Tapir Specialist Group, a unit of the IUCN Species Survival Commission, strives to conserve biological diversity by stimulating, developing, and conducting practical programs to study, save, restore, and manage the four species of tapir and their remaining habitats in Central and South America and Southeast Asia.


title: Tapir

text: The Baird's Tapir Project of Costa Rica, begun in 1994, is the longest ongoing tapir project in the world. It involves placing radio collars on tapirs in Costa Rica's Corcovado National Park to study their social systems and habitat preferences.


title: Tapir

text: The Lowland Lowland Tapir Conservation Initiative is a conservation and research organization founded by Patrícia Medici, focused on tapir conservation in Brazil.


title: Tapir

text: Tapirs are generally shy, but when scared they can defend themselves with their very powerful jaws. In 1998, a zookeeper in Oklahoma City was mauled and had an arm severed after opening the door to a female tapir's enclosure to push food inside (the tapir's two-month-old baby also occupied the cage at the time). In 2006, Carlos Manuel Rodriguez Echandi (who was then the Costa Rican Environmental Minister) became lost in the Corcovado National Park and was found by a search party with a "nasty bite" from a wild tapir. In 2013, a two-year-old girl suffered stomach and arm injuries after being mauled by a South American tapir in Dublin Zoo during a supervised experience in the tapir enclosure. Dublin Zoo pleaded guilty to breaching health and safety regulations and was ordered to pay €5,000 to charity. However, such examples are rare; for the most part, tapirs are likely to avoid confrontation in favour of running from predators, hiding, or, if possible, submerging themselves in nearby water until a threat is gone.


title: Tapir

text: Frank Buck wrote about an attack by a tapir in 1926, which he described in his book, "Bring 'Em Back Alive".


title: Tapir

text: Tapirs feature in the folklore of several cultures around the world. In Japan, tapirs are associated with the mythological Baku, believed to ward off nightmares. In South America, tapirs are associated with the creation of the earth.


title: Manifold

text: In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an formula_1-dimensional manifold, or "formula_1-manifold" for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of formula_1-dimensional Euclidean space.


title: Manifold

text: One-dimensional manifolds include lines and circles, but not lemniscates. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane.


title: Manifold

text: The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. The concept has applications in computer-graphics given the need to associate pictures with coordinates (e.g. CT scans).


title: Manifold

text: Manifolds can be equipped with additional structure. One important class of manifolds are differentiable manifolds; their differentiable structure allows calculus to be done. A Riemannian metric on a manifold allows distances and angles to be measured. Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model spacetime in general relativity.


title: Manifold

text: After a line, a circle is the simplest example of a topological manifold. Topology ignores bending, so a small piece of a circle is treated the same as a small piece of a line. Considering, for instance, the top part of the unit circle, "x" + "y" = 1, where the "y"-coordinate is positive (indicated by the yellow arc in "Figure 1"). Any point of this arc can be uniquely described by its "x"-coordinate. So, projection onto the first coordinate is a continuous and invertible mapping from the upper arc to the open interval (−1, 1):


title: Manifold

text: Such functions along with the open regions they map are called "charts". Similarly, there are charts for the bottom (red), left (blue), and right (green) parts of the circle:


title: Manifold

text: The top and right charts, formula_6 and formula_7 respectively, overlap in their domain: their intersection lies in the quarter of the circle where both formula_8 and formula_9-coordinates are positive. Both map this part into the interval formula_10, though differently. Thus a function formula_11 can be constructed, which takes values from the co-domain of formula_6 back to the circle using the inverse, followed by formula_7 back to the interval. For any number "a" in formula_10, then:


title: Manifold

text: The top, bottom, left, and right charts do not form the only possible atlas. Charts need not be geometric projections, and the number of charts is a matter of choice. Consider the charts


title: Manifold

text: Here "s" is the slope of the line through the point at coordinates ("x", "y") and the fixed pivot point (−1, 0); similarly, "t" is the opposite of the slope of the line through the points at coordinates ("x", "y") and (+1, 0). The inverse mapping from "s" to ("x", "y") is given by


title: Manifold

text: It can be confirmed that "x" + "y" = 1 for all values of "s" and "t". These two charts provide a second atlas for the circle, with the transition map


title: Manifold

text: (that is, one has this relation between "s" and "t" for every point where "s" and "t" are both nonzero).


title: Manifold

text: Each chart omits a single point, either (−1, 0) for "s" or (+1, 0) for "t", so neither chart alone is sufficient to cover the whole circle. It can be proved that it is not possible to cover the full circle with a single chart. For example, although it is possible to construct a circle from a single line interval by overlapping and "gluing" the ends, this does not produce a chart; a portion of the circle will be mapped to both ends at once, losing invertibility.


title: Manifold

text: may be covered by an atlas of six charts: the plane divides the sphere into two half spheres ( and ), which may both be mapped on the disc by the projection on the plane of coordinates. This provides two charts; the four other charts are provided by a similar construction with the two other coordinate planes.


title: Manifold

text: As with the circle, one may define one chart that covers the whole sphere excluding one point. Thus two charts are sufficient, but the sphere cannot be covered by a single chart.


title: Manifold

text: This example is historically significant, as it has motivated the terminology; it became apparent that the whole surface of the Earth cannot have a plane representation consisting of a single map (also called "chart", see nautical chart), and therefore one needs atlases for covering the whole Earth surface.


title: Manifold

text: Manifolds need not be closed; thus a line segment without its end points is a manifold. They are never countable, unless the dimension of the manifold is 0. Putting these freedoms together, other examples of manifolds are a parabola, a hyperbola, and the locus of points on a cubic curve (a closed loop piece and an open, infinite piece).


title: Manifold

text: However, excluded are examples like two touching circles that share a point to form a figure-8; at the shared point, a satisfactory chart cannot be created. Even with the bending allowed by topology, the vicinity of the shared point looks like a "+", not a line. A "+" is not homeomorphic to a line segment, since deleting the center point from the "+" gives a space with four components (i.e. pieces), whereas deleting a point from a line segment gives a space with at most two pieces; topological operations always preserve the number of pieces.


title: Manifold

text: There are many different kinds of manifolds. In geometry and topology, all manifolds are topological manifolds, possibly with additional structure. A manifold can be constructed by giving a collection of coordinate charts, that is, a covering by open sets with homeomorphisms to a Euclidean space, and patching functions: homeomorphisms from one region of Euclidean space to another region if they correspond to the same part of the manifold in two different coordinate charts. A manifold can be given additional structure if the patching functions satisfy axioms beyond continuity. For instance, differentiable manifolds have homeomorphisms on overlapping neighborhoods diffeomorphic with each other, so that the manifold has a well-defined set of functions which are differentiable in each neighborhood, thus differentiable on the manifold as a whole.


title: Manifold

text: Formally, a (topological) manifold is a second countable Hausdorff space that is locally homeomorphic to Euclidean space.


title: Manifold

text: "Second countable" and "Hausdorff" are point-set conditions; "second countable" excludes spaces which are in some sense 'too large' such as the long line, while "Hausdorff" excludes spaces such as "the line with two origins" (these generalizations of manifolds are discussed in non-Hausdorff manifolds).


title: Manifold

text: "Locally homeomorphic" to Euclidean space means that every point has a neighborhood homeomorphic to an open Euclidean "n"-ball,


title: Manifold

text: More precisely, locally homeomorphic here means that each point "m" in the manifold "M" has an open neighborhood homeomorphic to an open neighborhood in Euclidean space. However, given such a homeomorphism, the pre-image of an formula_21-ball gives a homeomorphism between the unit ball and a smaller neighborhood of "m", so this is no loss of generality. For topological or differentiable manifolds, one can also ask that every point have a neighborhood homeomorphic to all of Euclidean space (as this is diffeomorphic to the unit ball), but this cannot be done for complex manifolds, as the complex unit ball is not holomorphic to complex space.


title: Manifold

text: Generally manifolds are taken to have a fixed dimension (the space must be locally homeomorphic to a fixed "n"-ball), and such a space is called an "n"-manifold; however, some authors admit manifolds where different points can have different dimensions. If a manifold has a fixed dimension, it is called a pure manifold. For example, the (surface of a) sphere has a constant dimension of 2 and is therefore a pure manifold whereas the disjoint union of a sphere and a line in three-dimensional space is "not" a pure manifold. Since dimension is a local invariant (i.e. the map sending each point to the dimension of its neighbourhood over which a chart is defined, is locally constant), each connected component has a fixed dimension.


title: Manifold

text: Sheaf-theoretically, a manifold is a locally ringed space, whose structure sheaf is locally isomorphic to the sheaf of continuous (or differentiable, or complex-analytic, etc.) functions on Euclidean space. This definition is mostly used when discussing analytic manifolds in algebraic geometry.


title: Manifold

text: The spherical Earth is navigated using flat maps or charts, collected in an atlas. Similarly, a differentiable manifold can be described using mathematical maps, called "coordinate charts", collected in a mathematical "atlas". It is not generally possible to describe a manifold with just one chart, because the global structure of the manifold is different from the simple structure of the charts. For example, no single flat map can represent the entire Earth without separation of adjacent features across the map's boundaries or duplication of coverage. When a manifold is constructed from multiple overlapping charts, the regions where they overlap carry information essential to understanding the global structure.


title: Manifold

text: A "coordinate map", a "coordinate chart", or simply a "chart", of a manifold is an invertible map between a subset of the manifold and a simple space such that both the map and its inverse preserve the desired structure. For a topological manifold, the simple space is a subset of some Euclidean space formula_22 and interest focuses on the topological structure. This structure is preserved by homeomorphisms, invertible maps that are continuous in both directions.


title: Manifold

text: In the case of a differentiable manifold, a set of "charts" called an "atlas" allows us to do calculus on manifolds. Polar coordinates, for example, form a chart for the plane formula_23 minus the positive "x"-axis and the origin. Another example of a chart is the map χ mentioned above, a chart for the circle.


title: Manifold

text: The description of most manifolds requires more than one chart. A specific collection of charts which covers a manifold is called an "atlas". An atlas is not unique as all manifolds can be covered in multiple ways using different combinations of charts. Two atlases are said to be equivalent if their union is also an atlas.


title: Manifold

text: The atlas containing all possible charts consistent with a given atlas is called the "maximal atlas" (i.e. an equivalence class containing that given atlas). Unlike an ordinary atlas, the maximal atlas of a given manifold is unique. Though useful for definitions, it is an abstract object and not used directly (e.g. in calculations).


title: Manifold

text: Charts in an atlas may overlap and a single point of a manifold may be represented in several charts. If two charts overlap, parts of them represent the same region of the manifold, just as a map of Europe and a map of Russia may both contain Moscow. Given two overlapping charts, a "transition function" can be defined which goes from an open ball in formula_22 to the manifold and then back to another (or perhaps the same) open ball in formula_22. The resultant map, like the map "T" in the circle example above, is called a "change of coordinates", a "coordinate transformation", a "transition function", or a "transition map".


title: Manifold

text: An atlas can also be used to define additional structure on the manifold. The structure is first defined on each chart separately. If all transition maps are compatible with this structure, the structure transfers to the manifold.


title: Manifold

text: This is the standard way differentiable manifolds are defined. If the transition functions of an atlas for a topological manifold preserve the natural differential structure of formula_22 (that is, if they are diffeomorphisms), the differential structure transfers to the manifold and turns it into a differentiable manifold. Complex manifolds are introduced in an analogous way by requiring that the transition functions of an atlas are holomorphic functions. For symplectic manifolds, the transition functions must be symplectomorphisms.


title: Manifold

text: The structure on the manifold depends on the atlas, but sometimes different atlases can be said to give rise to the same structure. Such atlases are called "compatible".


title: Manifold

text: A manifold with boundary is a manifold with an edge. For example, a sheet of paper is a 2-manifold with a 1-dimensional boundary. The boundary of an "n"-manifold with boundary is an -manifold. A disk (circle plus interior) is a 2-manifold with boundary. Its boundary is a circle, a 1-manifold. A square with interior is also a 2-manifold with boundary. A ball (sphere plus interior) is a 3-manifold with boundary. Its boundary is a sphere, a 2-manifold. (Do not confuse with Boundary (topology)).


title: Manifold

text: In technical language, a manifold with boundary is a space containing both interior points and boundary points. Every interior point has a neighborhood homeomorphic to the open "n"-ball Every boundary point has a neighborhood homeomorphic to the "half" "n"-ball . The homeomorphism must send each boundary point to a point with "x" = 0.


title: Manifold

text: Let "M" be a manifold with boundary. The interior of "M", denoted Int "M", is the set of points in "M" which have neighborhoods homeomorphic to an open subset of formula_22. The boundary of "M", denoted ∂"M", is the complement of Int"M" in "M". The boundary points can be characterized as those points which land on the boundary hyperplane of formula_28 under some coordinate chart.


title: Manifold

text: If "M" is a manifold with boundary of dimension "n", then Int"M" is a manifold (without boundary) of dimension "n" and ∂"M" is a manifold (without boundary) of dimension .


title: Manifold

text: A single manifold can be constructed in different ways, each stressing a different aspect of the manifold, thereby leading to a slightly different viewpoint.


title: Manifold

text: Perhaps the simplest way to construct a manifold is the one used in the example above of the circle. First, a subset of formula_23 is identified, and then an atlas covering this subset is constructed. The concept of "manifold" grew historically from constructions like this. Here is another example, applying this method to the construction of a sphere:


title: Manifold

text: A sphere can be treated in almost the same way as the circle. In mathematics a sphere is just the surface (not the solid interior), which can be defined as a subset of formula_30:


title: Manifold

text: The sphere is two-dimensional, so each chart will map part of the sphere to an open subset of formula_23. Consider the northern hemisphere, which is the part with positive "z" coordinate (coloured red in the picture on the right). The function defined by


title: Manifold

text: maps the northern hemisphere to the open unit disc by projecting it on the ("x", "y") plane. A similar chart exists for the southern hemisphere. Together with two charts projecting on the ("x", "z") plane and two charts projecting on the ("y", "z") plane, an atlas of six charts is obtained which covers the entire sphere.


title: Manifold

text: A manifold can be constructed by gluing together pieces in a consistent manner, making them into overlapping charts. This construction is possible for any manifold and hence it is often used as a characterisation, especially for differentiable and Riemannian manifolds. It focuses on an atlas, as the patches naturally provide charts, and since there is no exterior space involved it leads to an intrinsic view of the manifold.


title: Manifold

text: The manifold is constructed by specifying an atlas, which is itself defined by transition maps. A point of the manifold is therefore an equivalence class of points which are mapped to each other by transition maps. Charts map equivalence classes to points of a single patch. There are usually strong demands on the consistency of the transition maps. For topological manifolds they are required to be homeomorphisms; if they are also diffeomorphisms, the resulting manifold is a differentiable manifold.


title: Manifold

text: This can be illustrated with the transition map "t" = ⁄ from the second half of the circle example. Start with two copies of the line. Use the coordinate "s" for the first copy, and "t" for the second copy. Now, glue both copies together by identifying the point "t" on the second copy with the point "s" = ⁄ on the first copy (the points "t" = 0 and "s" = 0 are not identified with any point on the first and second copy, respectively). This gives a circle.


title: Manifold

text: The first construction and this construction are very similar, but represent rather different points of view. In the first construction, the manifold is seen as embedded in some Euclidean space. This is the "extrinsic view". When a manifold is viewed in this way, it is easy to use intuition from Euclidean spaces to define additional structure. For example, in a Euclidean space, it is always clear whether a vector at some point is tangential or normal to some surface through that point.


title: Manifold

text: The patchwork construction does not use any embedding, but simply views the manifold as a topological space by itself. This abstract point of view is called the "intrinsic view". It can make it harder to imagine what a tangent vector might be, and there is no intrinsic notion of a normal bundle, but instead there is an intrinsic stable normal bundle.


title: Manifold

text: The "n"-sphere S is a generalisation of the idea of a circle (1-sphere) and sphere (2-sphere) to higher dimensions. An "n"-sphere S can be constructed by gluing together two copies of formula_22. The transition map between them is inversion in a sphere, defined as


title: Manifold

text: This function is its own inverse and thus can be used in both directions. As the transition map is a smooth function, this atlas defines a smooth manifold.


title: Manifold

text: It is possible to define different points of a manifold to be same. This can be visualized as gluing these points together in a single point, forming a quotient space. There is, however, no reason to expect such quotient spaces to be manifolds. Among the possible quotient spaces that are not necessarily manifolds, orbifolds and CW complexes are considered to be relatively well-behaved. An example of a quotient space of a manifold that is also a manifold is the real projective space, identified as a quotient space of the corresponding sphere.


title: Manifold

text: One method of identifying points (gluing them together) is through a right (or left) action of a group, which acts on the manifold. Two points are identified if one is moved onto the other by some group element. If "M" is the manifold and "G" is the group, the resulting quotient space is denoted by "M" / "G" (or "G" \ "M").


title: Manifold

text: Manifolds which can be constructed by identifying points include tori and real projective spaces (starting with a plane and a sphere, respectively).


title: Manifold

text: Two manifolds with boundaries can be glued together along a boundary. If this is done the right way, the result is also a manifold. Similarly, two boundaries of a single manifold can be glued together.


title: Manifold

text: Formally, the gluing is defined by a bijection between the two boundaries. Two points are identified when they are mapped onto each other. For a topological manifold, this bijection should be a homeomorphism, otherwise the result will not be a topological manifold. Similarly, for a differentiable manifold, it has to be a diffeomorphism. For other manifolds, other structures should be preserved.


title: Manifold

text: A finite cylinder may be constructed as a manifold by starting with a strip [0,1] × [0,1] and gluing a pair of opposite edges on the boundary by a suitable diffeomorphism. A projective plane may be obtained by gluing a sphere with a hole in it to a Möbius strip along their respective circular boundaries.


title: Manifold

text: The dimension of the product manifold is the sum of the dimensions of its factors. Its topology is the product topology, and a Cartesian product of charts is a chart for the product manifold. Thus, an atlas for the product manifold can be constructed using atlases for its factors. If these atlases define a differential structure on the factors, the corresponding atlas defines a differential structure on the product manifold. The same is true for any other structure defined on the factors. If one of the factors has a boundary, the product manifold also has a boundary. Cartesian products may be used to construct tori and finite cylinders, for example, as S × S and S × [0,1], respectively.


title: Manifold

text: The study of manifolds combines many important areas of mathematics: it generalizes concepts such as curves and surfaces as well as ideas from linear algebra and topology.


title: Manifold

text: Non-Euclidean geometry considers spaces where Euclid's parallel postulate fails. Saccheri first studied such geometries in 1733, but sought only to disprove them. Gauss, Bolyai and Lobachevsky independently discovered them 100 years later. Their research uncovered two types of spaces whose geometric structures differ from that of classical Euclidean space; these gave rise to hyperbolic geometry and elliptic geometry. In the modern theory of manifolds, these notions correspond to Riemannian manifolds with constant negative and positive curvature, respectively.


title: Manifold

text: Carl Friedrich Gauss may have been the first to consider abstract spaces as mathematical objects in their own right. His theorema egregium gives a method for computing the curvature of a surface without considering the ambient space in which the surface lies. Such a surface would, in modern terminology, be called a manifold; and in modern terms, the theorem proved that the curvature of the surface is an intrinsic property. Manifold theory has come to focus exclusively on these intrinsic properties (or invariants), while largely ignoring the extrinsic properties of the ambient space.


title: Manifold

text: Another, more topological example of an intrinsic property of a manifold is its Euler characteristic. Leonhard Euler showed that for a convex polytope in the three-dimensional Euclidean space with "V" vertices (or corners), "E" edges, and "F" faces,formula_36The same formula will hold if we project the vertices and edges of the polytope onto a sphere, creating a topological map with "V" vertices, "E" edges, and "F" faces, and in fact, will remain true for any spherical map, even if it does not arise from any convex polytope. Thus 2 is a topological invariant of the sphere, called its Euler characteristic. On the other hand, a torus can be sliced open by its 'parallel' and 'meridian' circles, creating a map with "V" = 1 vertex, "E" = 2 edges, and "F" = 1 face. Thus the Euler characteristic of the torus is 1 − 2 + 1 = 0. The Euler characteristic of other surfaces is a useful topological invariant, which can be extended to higher dimensions using Betti numbers. In the mid nineteenth century, the Gauss–Bonnet theorem linked the Euler characteristic to the Gaussian curvature.


title: Manifold

text: Investigations of Niels Henrik Abel and Carl Gustav Jacobi on inversion of elliptic integrals in the first half of 19th century led them to consider special types of complex manifolds, now known as Jacobians. Bernhard Riemann further contributed to their theory, clarifying the geometric meaning of the process of analytic continuation of functions of complex variables.


title: Manifold

text: Another important source of manifolds in 19th century mathematics was analytical mechanics, as developed by Siméon Poisson, Jacobi, and William Rowan Hamilton. The possible states of a mechanical system are thought to be points of an abstract space, phase space in Lagrangian and Hamiltonian formalisms of classical mechanics. This space is, in fact, a high-dimensional manifold, whose dimension corresponds to the degrees of freedom of the system and where the points are specified by their generalized coordinates. For an unconstrained movement of free particles the manifold is equivalent to the Euclidean space, but various conservation laws constrain it to more complicated formations, e.g. Liouville tori. The theory of a rotating solid body, developed in the 18th century by Leonhard Euler and Joseph-Louis Lagrange, gives another example where the manifold is nontrivial. Geometrical and topological aspects of classical mechanics were emphasized by Henri Poincaré, one of the founders of topology.


title: Manifold

text: Riemann was the first one to do extensive work generalizing the idea of a surface to higher dimensions. The name "manifold" comes from Riemann's original German term, "Mannigfaltigkeit", which William Kingdon Clifford translated as "manifoldness". In his Göttingen inaugural lecture, Riemann described the set of all possible values of a variable with certain constraints as a "Mannigfaltigkeit", because the variable can have "many" values. He distinguishes between "stetige Mannigfaltigkeit" and "diskrete" "Mannigfaltigkeit" ("continuous manifoldness" and "discontinuous manifoldness"), depending on whether the value changes continuously or not. As continuous examples, Riemann refers to not only colors and the locations of objects in space, but also the possible shapes of a spatial figure. Using induction, Riemann constructs an "n-fach ausgedehnte Mannigfaltigkeit" ("n times extended manifoldness" or "n-dimensional manifoldness") as a continuous stack of (n−1) dimensional manifoldnesses. Riemann's intuitive notion of a "Mannigfaltigkeit" evolved into what is today formalized as a manifold. Riemannian manifolds and Riemann surfaces are named after Riemann.


title: Manifold

text: In his very influential paper, Analysis Situs, Henri Poincaré gave a definition of a differentiable manifold ("variété") which served as a precursor to the modern concept of a manifold.


title: Manifold

text: In the first section of Analysis Situs, Poincaré defines a manifold as the level set of a continuously differentiable function between Euclidean spaces that satisfies the nondegeneracy hypothesis of the implicit function theorem. In the third section, he begins by remarking that the graph of a continuously differentiable function is a manifold in the latter sense. He then proposes a new, more general, definition of manifold based on a 'chain of manifolds' ("une chaîne des variétés").


title: Manifold

text: Poincaré's notion of a "chain of manifolds" is a precursor to the modern notion of atlas. In particular, he considers two manifolds defined respectively as graphs of functions formula_37 and formula_38. If these manifolds overlap ("a une partie commune"), then he requires that the coordinates formula_39 depend continuously differentiably on the coordinates formula_40 and vice versa ("'...les formula_41 sont fonctions analytiques des formula_42 et inversement"'). In this way he introduces a precursor to the notion of a chart and of a transition map.


title: Manifold

text: For example, the unit circle in the plane can be thought of as the graph of the function formula_43 or else the function formula_44 in a neighborhood of every point except the points (1, 0) and (−1, 0); and in a neighborhood of those points, it can be thought of as the graph of, respectively, formula_45 and formula_46. The circle can be represented by a graph in the neighborhood of every point because the left hand side of its defining equation formula_47 has nonzero gradient at every point of the circle. By the implicit function theorem, every submanifold of Euclidean space is locally the graph of a function.


title: Manifold

text: Hermann Weyl gave an intrinsic definition for differentiable manifolds in his lecture course on Riemann surfaces in 1911–1912, opening the road to the general concept of a topological space that followed shortly. During the 1930s Hassler Whitney and others clarified the foundational aspects of the subject, and thus intuitions dating back to the latter half of the 19th century became precise, and developed through differential geometry and Lie group theory. Notably, the Whitney embedding theorem showed that the intrinsic definition in terms of charts was equivalent to Poincaré's definition in terms of subsets of Euclidean space.


title: Manifold

text: Two-dimensional manifolds, also known as a 2D "surfaces" embedded in our common 3D space, were considered by Riemann under the guise of Riemann surfaces, and rigorously classified in the beginning of the 20th century by Poul Heegaard and Max Dehn. Poincaré pioneered the study of three-dimensional manifolds and raised a fundamental question about them, today known as the Poincaré conjecture. After nearly a century, Grigori Perelman proved the Poincaré conjecture (see the Solution of the Poincaré conjecture). William Thurston's geometrization program, formulated in the 1970s, provided a far-reaching extension of the Poincaré conjecture to the general three-dimensional manifolds. Four-dimensional manifolds were brought to the forefront of mathematical research in the 1980s by Michael Freedman and in a different setting, by Simon Donaldson, who was motivated by the then recent progress in theoretical physics (Yang–Mills theory), where they serve as a substitute for ordinary 'flat' spacetime. Andrey Markov Jr. showed in 1960 that no algorithm exists for classifying four-dimensional manifolds. Important work on higher-dimensional manifolds, including analogues of the Poincaré conjecture, had been done earlier by René Thom, John Milnor, Stephen Smale and Sergei Novikov. A very pervasive and flexible technique underlying much work on the topology of manifolds is Morse theory.


title: Manifold

text: The simplest kind of manifold to define is the topological manifold, which looks locally like some "ordinary" Euclidean space formula_22. By definition, all manifolds are topological manifolds, so the phrase "topological manifold" is usually used to emphasize that a manifold lacks additional structure, or that only its topological properties are being considered. Formally, a topological manifold is a topological space locally homeomorphic to a Euclidean space. This means that every point has a neighbourhood for which there exists a homeomorphism (a bijective continuous function whose inverse is also continuous) mapping that neighbourhood to formula_22. These homeomorphisms are the charts of the manifold.


title: Manifold

text: A "topological" manifold looks locally like a Euclidean space in a rather weak manner: while for each individual chart it is possible to distinguish differentiable functions or measure distances and angles, merely by virtue of being a topological manifold a space does not have any "particular" and "consistent" choice of such concepts. In order to discuss such properties for a manifold, one needs to specify further structure and consider differentiable manifolds and Riemannian manifolds discussed below. In particular, the same underlying topological manifold can have several mutually incompatible classes of differentiable functions and an infinite number of ways to specify distances and angles.


title: Manifold

text: Usually additional technical assumptions on the topological space are made to exclude pathological cases. It is customary to require that the space be Hausdorff and second countable.


title: Manifold

text: The "dimension" of the manifold at a certain point is the dimension of the Euclidean space that the charts at that point map to (number "n" in the definition). All points in a connected manifold have the same dimension. Some authors require that all charts of a topological manifold map to Euclidean spaces of same dimension. In that case every topological manifold has a topological invariant, its dimension.


title: Manifold

text: For most applications, a special kind of topological manifold, namely, a differentiable manifold, is used. If the local charts on a manifold are compatible in a certain sense, one can define directions, tangent spaces, and differentiable functions on that manifold. In particular it is possible to use calculus on a differentiable manifold. Each point of an "n"-dimensional differentiable manifold has a tangent space. This is an "n"-dimensional Euclidean space consisting of the tangent vectors of the curves through the point.


title: Manifold

text: Two important classes of differentiable manifolds are smooth and analytic manifolds. For smooth manifolds the transition maps are smooth, that is, infinitely differentiable. Analytic manifolds are smooth manifolds with the additional condition that the transition maps are analytic (they can be expressed as power series). The sphere can be given analytic structure, as can most familiar curves and surfaces.


title: Manifold

text: A rectifiable set generalizes the idea of a piecewise smooth or rectifiable curve to higher dimensions; however, rectifiable sets are not in general manifolds.


title: Manifold

text: To measure distances and angles on manifolds, the manifold must be Riemannian. A "Riemannian manifold" is a differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. Given two tangent vectors and , the inner product gives a real number. The dot (or scalar) product is a typical example of an inner product. This allows one to define various notions such as length, angles, areas (or volumes), curvature and divergence of vector fields.


title: Manifold

text: All differentiable manifolds (of constant dimension) can be given the structure of a Riemannian manifold. The Euclidean space itself carries a natural structure of Riemannian manifold (the tangent spaces are naturally identified with the Euclidean space itself and carry the standard scalar product of the space). Many familiar curves and surfaces, including for example all -spheres, are specified as subspaces of a Euclidean space and inherit a metric from their embedding in it.


title: Manifold

text: A Finsler manifold allows the definition of distance but does not require the concept of angle; it is an analytic manifold in which each tangent space is equipped with a norm, ||·||, in a manner which varies smoothly from point to point. This norm can be extended to a metric, defining the length of a curve; but it cannot in general be used to define an inner product.


title: Manifold

text: Lie groups, named after Sophus Lie, are differentiable manifolds that carry also the structure of a group which is such that the group operations are defined by smooth maps.


title: Manifold

text: A Euclidean vector space with the group operation of vector addition is an example of a non-compact Lie group. A simple example of a compact Lie group is the circle: the group operation is simply rotation. This group, known as U(1), can be also characterised as the group of complex numbers of modulus 1 with multiplication as the group operation.


title: Manifold

text: Other examples of Lie groups include special groups of matrices, which are all subgroups of the general linear group, the group of "n" by "n" matrices with non-zero determinant. If the matrix entries are real numbers, this will be an "n"-dimensional disconnected manifold. The orthogonal groups, the symmetry groups of the sphere and hyperspheres, are "n"("n"−1)/2 dimensional manifolds, where "n"−1 is the dimension of the sphere. Further examples can be found in the table of Lie groups.


title: Manifold

text: Different notions of manifolds have different notions of classification and invariant; in this section we focus on smooth closed manifolds.


title: Manifold

text: The classification of smooth closed manifolds is well understood "in principle", except in dimension 4: in low dimensions (2 and 3) it is geometric, via the uniformization theorem and the solution of the Poincaré conjecture, and in high dimension (5 and above) it is algebraic, via surgery theory. This is a classification in principle: the general question of whether two smooth manifolds are diffeomorphic is not computable in general. Further, specific computations remain difficult, and there are many open questions.


title: Manifold

text: Orientable surfaces can be visualized, and their diffeomorphism classes enumerated, by genus. Given two orientable surfaces, one can determine if they are diffeomorphic by computing their respective genera and comparing: they are diffeomorphic if and only if the genera are equal, so the genus forms a complete set of invariants.


title: Manifold

text: This is much harder in higher dimensions: higher-dimensional manifolds cannot be directly visualized (though visual intuition is useful in understanding them), nor can their diffeomorphism classes be enumerated, nor can one in general determine if two different descriptions of a higher-dimensional manifold refer to the same object.


title: Manifold

text: However, one can determine if two manifolds are "different" if there is some intrinsic characteristic that differentiates them. Such criteria are commonly referred to as invariants, because, while they may be defined in terms of some presentation (such as the genus in terms of a triangulation), they are the same relative to all possible descriptions of a particular manifold: they are "invariant" under different descriptions.


title: Manifold

text: Naively, one could hope to develop an arsenal of invariant criteria that would definitively classify all manifolds up to isomorphism. Unfortunately, it is known that for manifolds of dimension 4 and higher, no program exists that can decide whether two manifolds are diffeomorphic.


title: Manifold

text: Smooth manifolds have a rich set of invariants, coming from point-set topology, classic algebraic topology, and geometric topology. The most familiar invariants, which are visible for surfaces, are orientability (a normal invariant, also detected by homology) and genus (a homological invariant).


title: Manifold

text: Smooth closed manifolds have no local invariants (other than dimension), though geometric manifolds have local invariants, notably the curvature of a Riemannian manifold and the torsion of a manifold equipped with an affine connection. This distinction between local invariants and no local invariants is a common way to distinguish between geometry and topology. All invariants of a smooth closed manifold are thus global.


title: Manifold

text: Algebraic topology is a source of a number of important global invariant properties. Some key criteria include the "simply connected" property and orientability (see below). Indeed, several branches of mathematics, such as homology and homotopy theory, and the theory of characteristic classes were founded in order to study invariant properties of manifolds.


title: Manifold

text: In dimensions two and higher, a simple but important invariant criterion is the question of whether a manifold admits a meaningful orientation. Consider a topological manifold with charts mapping to formula_22. Given an ordered basis for formula_22, a chart causes its piece of the manifold to itself acquire a sense of ordering, which in 3-dimensions can be viewed as either right-handed or left-handed. Overlapping charts are not required to agree in their sense of ordering, which gives manifolds an important freedom. For some manifolds, like the sphere, charts can be chosen so that overlapping regions agree on their "handedness"; these are "orientable" manifolds. For others, this is impossible. The latter possibility is easy to overlook, because any closed surface embedded (without self-intersection) in three-dimensional space is orientable.


title: Manifold

text: Some illustrative examples of non-orientable manifolds include: (1) the Möbius strip, which is a manifold with boundary, (2) the Klein bottle, which must intersect itself in its 3-space representation, and (3) the real projective plane, which arises naturally in geometry.


title: Manifold

text: Begin with an infinite circular cylinder standing vertically, a manifold without boundary. Slice across it high and low to produce two circular boundaries, and the cylindrical strip between them. This is an orientable manifold with boundary, upon which "surgery" will be performed. Slice the strip open, so that it could unroll to become a rectangle, but keep a grasp on the cut ends. Twist one end 180°, making the inner surface face out, and glue the ends back together seamlessly. This results in a strip with a permanent half-twist: the Möbius strip. Its boundary is no longer a pair of circles, but (topologically) a single circle; and what was once its "inside" has merged with its "outside", so that it now has only a "single" side. Similarly to the Klein Bottle below, this two dimensional surface would need to intersect itself in two dimensions, but can easily be constructed in three or more dimensions.


title: Manifold

text: Take two Möbius strips; each has a single loop as a boundary. Straighten out those loops into circles, and let the strips distort into cross-caps. Gluing the circles together will produce a new, closed manifold without boundary, the Klein bottle. Closing the surface does nothing to improve the lack of orientability, it merely removes the boundary. Thus, the Klein bottle is a closed surface with no distinction between inside and outside. In three-dimensional space, a Klein bottle's surface must pass through itself. Building a Klein bottle which is not self-intersecting requires four or more dimensions of space.


title: Manifold

text: Begin with a sphere centered on the origin. Every line through the origin pierces the sphere in two opposite points called "antipodes". Although there is no way to do so physically, it is possible (by considering a quotient space) to mathematically merge each antipode pair into a single point. The closed surface so produced is the real projective plane, yet another non-orientable surface. It has a number of equivalent descriptions and constructions, but this route explains its name: all the points on any given line through the origin project to the same "point" on this "plane".


title: Manifold

text: For two dimensional manifolds a key invariant property is the genus, or "number of handles" present in a surface. A torus is a sphere with one handle, a double torus is a sphere with two handles, and so on. Indeed, it is possible to fully characterize compact, two-dimensional manifolds on the basis of genus and orientability. In higher-dimensional manifolds genus is replaced by the notion of Euler characteristic, and more generally Betti numbers and homology and cohomology.


title: Manifold

text: Just as there are various types of manifolds, there are various types of maps of manifolds. In addition to continuous functions and smooth functions generally, there are maps with special properties. In geometric topology a basic type are embeddings, of which knot theory is a central example, and generalizations such as immersions, submersions, covering spaces, and ramified covering spaces.


title: Manifold

text: In Riemannian geometry, one may ask for maps to preserve the Riemannian metric, leading to notions of isometric embeddings, isometric immersions, and Riemannian submersions; a basic result is the Nash embedding theorem.


title: Manifold

text: sometimes called regular functions or functionals, by analogy with algebraic geometry or linear algebra. These are of interest both in their own right, and to study the underlying manifold.


title: Manifold

text: In geometric topology, most commonly studied are Morse functions, which yield handlebody decompositions, while in mathematical analysis, one often studies solution to partial differential equations, an important example of which is harmonic analysis, where one studies harmonic functions: the kernel of the Laplace operator. This leads to such functions as the spherical harmonics, and to heat kernel methods of studying manifolds, such as hearing the shape of a drum and some proofs of the Atiyah–Singer index theorem.


title: Home Sweet Home Alone

text: Home Sweet Home Alone is a 2021 American Christmas comedy film directed by Dan Mazer, written by Mikey Day and Streeter Seidell, and starring Ellie Kemper, Rob Delaney, Archie Yates, Aisling Bea, Kenan Thompson, Pete Holmes, Ally Maki, and Chris Parnell. The sixth film in the "Home Alone" franchise, "Home Sweet Home Alone" was produced by 20th Century Studios as an original title for Disney+, the first 20th Century Studios film to be produced for the streaming service. The film was announced after The Walt Disney Company acquired 21st Century Fox and inherited the rights to the "Home Alone" franchise. The film centers on a boy who is left behind at the time when a married couple plan to get inside thinking he took a malformed doll that belonged to them.


title: Home Sweet Home Alone

text: Jeff and Pam McKenzie are trying to sell their house, but have not told their children, Abby and Chris. Jeff lost his job and Pam's salary is not enough to keep their home. To make matters worse, Jeff's obnoxious and successful brother Hunter, his wife, Mei, and their son Ollie have decided to stay with them for Christmas.


title: Home Sweet Home Alone

text: During an open house, Max Mercer and his mother Carol stop by to use the restroom. Max has a brief exchange with Jeff, during which the latter reveals a box of old dolls, including a malformed one with an upside down face. Carol explains to Jeff that dolls with unusual deformities are rare and worth a lot of money, with an estimated $200,000. As Max and Carol return home, the whole family is preparing to leave for Tokyo, Japan, for the holidays, with Carol leaving earlier than the rest. Max, annoyed with the fuss, settles in the garage-parked car to watch cartoons and accidentally falls asleep without knowing it. Unfortunately, Max's family didn't realize that their son was sleeping in the car parked in the garage and left him at home, thinking he was in another vehicle with one of the rest of the members.


title: Home Sweet Home Alone

text: Concerned about losing their home, Jeff goes to get the doll only to find it is missing. Believing that Max stole it, he locates the Mercer household the next day only to find the whole family hastily leaving. During the commotion, he overhears the security code and sees where the house key is hidden. Telling Pam about it, they agree to get the doll back at night.


title: Home Sweet Home Alone

text: Meanwhile, Max discovers the whole family has left and initially has fun, though he quickly grows bored and misses them. Meanwhile, Jeff and Pam arrive at the Mercer's and get inside. Overhearing them talk about getting an "ugly little boy", Max assumes they are talking about kidnapping and selling him. He attempts to scare them away by calling the police. Officer Buzz McCallister arrives, but Pam diverts him. Max fears if Officer McCallister realizes he is home alone, his parents may get arrested.


title: Home Sweet Home Alone

text: Carol discovers Max was left behind and buys a ticket to head back. The McKenzies go to church the next day and run into their realtor. He tells them there is a buyer, but they must decide by the end of the year, which puts extensive pressure on them. Max arrives and unknowingly converses with Jeff and Pam's son Chris who sympathetically gives him his water gun. The couple spot Max talking with someone and assume she is his grandmother. They resolve to break into the house once again while their family is still at church.


title: Home Sweet Home Alone

text: Sneaking around the back of the house, Jeff and Pam end up in the neighbor's backyard instead. Max overhears them once again, with Jeff agreeing to arrive dressed as Santa in an effort to fool him. Max responds by boobie-trapping the house to cause difficulty and prevent them from getting close to himself while Jeff and Pam wait for their family to fall asleep on Christmas Eve. The couple fall into Max's traps, during which they discover that Max did not steal the doll, but instead a can of soda. They clear up the misunderstanding, but learn Max is home alone and agree to let him stay with them until his mother returns.


title: Home Sweet Home Alone

text: As they explain the whole situation to their family, it turns out that Ollie stole the doll and managed to safely retrieve it, thus ensuring that the McKenzies can stay. Carol arrives to pick up Max. Having moved into their house only two months ago, Carol finds friends in the McKenzies and thanks them for taking care of Max.


title: Home Sweet Home Alone

text: One year later, the Mercers and McKenzies have Christmas dinner together. Jeff has gotten a new job and willingly gives Max the soda he craved the year prior.


title: Home Sweet Home Alone

text: On August 6, 2019, Disney CEO Bob Iger announced that a new film in the "Home Alone" franchise, titled simply "Home Alone", was in development. By October, Dan Mazer had entered negotiations to direct the film, with a script co-written by Mikey Day and Streeter Seidell. Hutch Parker and Dan Wilson served as producers.


title: Home Sweet Home Alone

text: In December 2019, Archie Yates, Rob Delaney, and Ellie Kemper were announced as the co-stars of the film. In July 2020, it was reported that Ally Maki, Kenan Thompson, Chris Parnell, Aisling Bea, Pete Holmes, Timothy Simons, and Mikey Day had joined the cast. In April 2020, it was reported that Macaulay Culkin, who played Kevin McCallister in the first two films, would reprise his role in a cameo; In October 2021, Culkin denied his involvement in the film. In August 2021, it was announced that Devin Ratray, who played Buzz McCallister in the first two films, would appear.


title: Home Sweet Home Alone

text: Principal photography began in February 2020, in Montreal, Quebec, Canada. In March, filming was halted due to the COVID-19 pandemic and industry restrictions worldwide. In November 2020, Disney announced that all its films that had been postponed by the coronavirus had resumed filming, and in some cases completed principal photography. The film was released digitally on Disney+ on November 12, 2021.


title: Home Sweet Home Alone

text: The score of the film was composed and conducted by John Debney, which incorporates John Williams's themes from the first two films. Two tracks in the score, "He's Hurting Me", and "Who Raised This Monster?", included a reference to Alan Silvestri's theme from the film, "Mouse Hunt".


title: Home Sweet Home Alone

text: The first trailer was released on October 12, 2021. It was met with negative responses by fans, and received more than 79,000 dislikes on YouTube in the first three days of release.


title: Home Sweet Home Alone

text: Aisling Bea, who is Irish, received criticism for using an English accent in the film. She said in an interview that "people were trying to connect it to some form of oppression, because it was American people getting me to do an English accent". She said she had used many accents in her career and dismissed the criticism as "sort of what Twitter was created for: people to complain about things that don't matter".


title: Home Sweet Home Alone

text: The site's critical consensus reads, "Nobody's "Home"." On Metacritic, the film has a weighted average score of 35 out of 100, based on 16 critics, indicating "generally unfavorable reviews".


title: Home Sweet Home Alone

text: Benjamin Lee of "The Guardian" gave the film 3 out of 5 stars and wrote: ""Home Sweet Home Alone" is a surprisingly entertaining, if wholly unnecessary, sequel, a tangerine where we expected to find a lump of coal." Jennifer Green of "Common Sense Media" also gave it 3 out of 5 stars and wrote: "By this point, after no fewer than five previous features in the franchise, "Home Sweet Home Alone" might not have much new to offer. But the characters here have a wholesomeness to them that was missing in some of the earlier films." Aditya Mani Jha of "Firstpost" rated the film 3 out of 5 stars and wrote: "Overall, "Home Sweet Home Alone" is a competently mounted but unambitious film. At times, it plays like a showreel from its source material, and the makers are okay with it being that way. This makes it efficient in the blockbuster sense, but even its keenest backers will find it difficult to remember too many of the film’s all-new scenarios or one-liners after the end credits roll." Adam Graham of "The Detroit News" gave it a grade of B- and wrote: ""Home Sweet Home Alone" holds its own, a new spin on an old tale that keeps the spirit of the original alive."


title: Home Sweet Home Alone

text: CNN's Brian Lowry wrote that the film "is a very odd duck -- a movie that basically replicates the three-decades-old "Home Alone" template, but in a way that feels slightly weird and ill-conceived." Tim Robey of "The Daily Telegraph" gave the film 2 out of 5 stars, writing: "Don't expect a Christmas miracle, then: it's more like a box of Quality Street that got heavily plundered before the wrapping went on." Clarisse Loughrey of "The Independent" gave the film 1 out of 5 stars, writing: ""Home Sweet Home Alone" decides, somewhat arbitrarily, to switch everything up – the kid is now an utter arsehole while the burglars are sympathetic, which results in the audience rooting for criminals while really just hoping that everyone would quit it and go to sleep." Kevin Maher of "The Times" also gave the film 1 out of 5 stars, writing that "everything that happens before the mayhem — the burglary begins after a full hour of padding — is excruciatingly forced." Jake Wilson of "The Age" also gave 1 out of 5 stars, writing: "This is less an actual movie than a lazy approximation, half-heartedly monkeying with formula while showing no understanding of why that formula ever worked at all."


title: Home Sweet Home Alone

text: Chris Columbus, the original director of the first two "Home Alone" films was also critical towards the decision in making a sixth movie prior to the release of the film. On the topic of movie remakes, he said, "In this version of Hollywood that we live in, everybody is remaking everything, and rebooting everything. I mean, there's a "Home Alone" reboot coming out... What's the point? The movie exists, let's just live with the movie that existed. There's no point in any of us remaking "The Wizard of Oz", there's no point in any of us remaking the classic films. Make something original, because we need more original material."


title: Ndamukong Suh

text: Ndamukong Ngwa Suh ( , born January 6, 1987) is an American football defensive tackle for the Philadelphia Eagles of the National Football League (NFL). He played college football at Nebraska, where he earned All-American honors, and was drafted by the Detroit Lions second overall in the 2010 NFL Draft. He also played for the Miami Dolphins, Los Angeles Rams, and Tampa Bay Buccaneers. He has played in two Super Bowls: Super Bowl LIII with the Rams and Super Bowl LV with the Buccaneers.


title: Ndamukong Suh

text: As a college senior, Suh became one of the most decorated players in college football history. He won numerous awards including the Associated Press College Football Player of the Year Award, Bronko Nagurski Trophy, Chuck Bednarik Award, Lombardi Award, and Outland Trophy, and was recognized as a unanimous All-American.


title: Ndamukong Suh

text: During his time with the Lions, Suh was selected to 4 All-Pro and Pro Bowl teams and was awarded Defensive Rookie of the Year in 2010. In 2015, Suh became the highest-paid defensive player in NFL history, having signed a six-year contract with the Miami Dolphins, worth in excess of $114 million, with nearly $60 million fully guaranteed; he was released after just three seasons. Early in his career, Suh was criticized for his aggressive style of play and lack of sportsmanship. Suh was one of four defensive tackles named to the NFL 2010's All-Decade Team.


title: Ndamukong Suh

text: Suh was born in Portland, Oregon. He attended Grant High School in Portland, where he was a four-sport star in football, basketball, soccer and track and field. He played as a two-way lineman for the Grant Generals. Suh earned first-team All-PIL honors on both offense and defense as a junior and was an honorable-mention All-state pick. In his senior year, he collected 65 tackles, including 10 sacks and recovered four fumbles, which earned him "Parade" magazine high school All-America honors, the 2004 Portland Interscholastic League Defensive Player of the Year, and a Class 4A first-team all-state selection. He also played in the 2005 U.S. Army All-American Bowl. In basketball, he earned honorable-mention All-League honors as a junior and senior.


title: Ndamukong Suh

text: Also a track & field athlete, Suh was one of the state's top performers in the shot put. He was the district shot put champion in 2004, and won the OSAA Class 4A shot put title in 2005 with a school-record throw of 18.71 meters (61 ft, 4 in). For his all-around athletic accomplishments, Suh was a finalist for the "Portland Tribune"’s Athlete of the Year.


title: Ndamukong Suh

text: Considered a four-star recruit by "Rivals.com", Suh was the sixth ranked defensive tackle in the United States. Recruited by many, Suh took official visits to Nebraska, Mississippi State, Oregon State, Miami (FL), and California, before committing to the Nebraska Cornhuskers on January 20, 2005.


title: Ndamukong Suh

text: Suh attended the University of Nebraska, where he played for the Nebraska Cornhuskers football team from 2005 to 2009. As a true freshman in 2005, Suh played in the first two games and had an assisted tackle against Wake Forest before missing the remainder of the season after undergoing knee surgery. He received a medical redshirt.


title: Ndamukong Suh

text: In 2006, Suh played in all 14 games as a backup defensive lineman and earned freshman All-Big 12 honors from "The Sporting News". Despite coming off the bench, he finished the year with 19 total tackles, and ranked among the team leaders in tackles for loss (8) and quarterback sacks (3½). In his sophomore season, Suh started in 11 of the Cornhuskers' 12 games, and recorded 34 total tackles on the season.


title: Ndamukong Suh

text: As a junior in 2008, Suh recorded a team-high 76 tackles, 7.5 sacks, 19 tackles for loss, 2 interceptions (both returned for touchdowns), and a touchdown reception while playing fullback. He became the first Nebraska defensive lineman to lead the team in tackles since 1973. Suh earned First-team All-Big 12 honors in 2008, the first Nebraska interior defensive lineman to earn those honors since Steve Warren in 1999. He was also an honorable mention All-American.


title: Ndamukong Suh

text: In 2009, Suh registered 85 tackles, 12 quarterback sacks, 28 quarterback hurries, 20.5 tackles for loss, 10 pass breakups, 3 blocked kicks, and 1 interception. Suh had 12 tackles (seven for losses, a school single-game record) and 4½ sacks in a 13–12 loss to the Texas Longhorns in the Big 12 Championship Game, for which he received game MVP honors. He helped Huskers defense rank first nationally in scoring defense (10.4 ppg), tied for second in total sacks (44), first in pass efficiency defense (87.3), seventh in total defense (272.0 ypg), ninth in rushing defense (93.1 ypg) and 18th in passing defense (178.9 ypg). He also played all four quarters versus Arizona in the 2009 Holiday Bowl, helping Nebraska record the first shutout in the Holiday Bowl's 32-year history, as well as the first shutout in school bowl history. He earned unanimous first-team All-Big 12 honors and the Big-12 Defensive Player of the Year. Nationally, he was named the Associated Press College Football Player of the Year, the recipient of the Bill Willis Trophy, Bronko Nagurski Trophy, Chuck Bednarik Award, Lombardi Award and Outland Trophy, and was recognized as a unanimous first-team All-American. He was also a finalist for the Lott Trophy, Walter Camp Award, and Heisman Trophy.


title: Ndamukong Suh

text: On December 3, 2009, Suh was named as one of five finalists for the Walter Camp Award. On November 24, 2009, Suh was named one of three finalists for the 2009 Outland Trophy, alongside Mike Iupati and Russell Okung. On November 10, Suh was selected one of four finalists for the 2009 Lombardi Award, the first Cornhusker to receive this honor since Dominic Raiola in 2000. In October 2009, Suh was named to "The Sporting News" and "CBS Sports" midseason All-American team. Suh began season at No. 3 on "Rivals.com′s" preseason defensive tackle power ranking. He was also named to the 2009 Outland Trophy watch list.


title: Ndamukong Suh

text: On December 7, 2009, Suh was named a finalist for the Heisman Trophy. Later that evening, Suh was named the 2009 Bronko Nagurski Trophy winner as the top defensive player in the nation. CBS Sportsline also named Suh their Defensive Player of the Year. The Touchdown Club of Columbus named Suh the winner of the Bill Willis Trophy on December 9, 2009. That same evening Suh won the Lombardi Award for the top collegiate lineman or linebacker. On December 11, at the ESPN College Football Awards show, Suh was selected as the winner of the Chuck Bednarik Award as the nation's best defensive player and took home the Outland Trophy for the best interior lineman. Suh finished fourth in the Heisman race, accumulating 815 points, the most by a fourth-place finisher for the Heisman Trophy in its history. He was also one of four unanimous selections to the AP First-team All-America in 2009. Suh was named the 2009 AP Player of Year, becoming the first defensive player to receive the award in its history.


title: Ndamukong Suh

text: Suh was widely considered to be one of the best prospects available in the draft. "ESPN.com's" draft analyst Mel Kiper, Jr., described Suh as "maybe the most dominating defensive tackle I've seen in 32 years" and projected him to go #1 overall. Suh was seen as an ideal fit at either defensive tackle in a 4–3 defense or as a defensive end in a 3–4 defense.


title: Ndamukong Suh

text: For off-the-field marketing activities, Suh signed with The Agency Sports Management & Marketing, where Russ Spielman served as lead agent. At the NFL Combine, Suh bench pressed 225 lbs 32 times and had a 35½ inch vertical leap, the highest for a defensive tackle since Al Lucas (36 in) in 2000.


title: Ndamukong Suh

text: Before the NFL draft, Suh signed with Maximum Sports Management and agent Roosevelt Barnes. This caused moderate concern for many teams who were hoping to draft him, as this was the same agent who represented Michael Crabtree. Crabtree was the longest 2009 NFL Draft first round contract hold out, waiting over six weeks into the NFL season before signing with the San Francisco 49ers. Crabtree had even threatened to re-enter the 2010 NFL Draft.


title: Ndamukong Suh

text: Despite the concerns, Suh was selected second overall in the 2010 NFL Draft by the Detroit Lions, becoming the first defensive lineman selected by the team since Luther Elliss in 1995. Suh was the highest selected Cornhusker defender since Neil Smith in 1988.


title: Ndamukong Suh

text: On August 3, Suh agreed to a five-year, $68 million contract with the Lions, including $40 million guaranteed. On September 12, Suh had his first sack against Chicago Bears quarterback Jay Cutler. On October 10, during a game against the St. Louis Rams, he recorded his first and only, as of 2021, career interception off Sam Bradford. Suh scored the first touchdown of his NFL career against the Washington Redskins on October 31, 2010, on a recovery of a Rex Grossman fumble.


title: Ndamukong Suh

text: After an injury to Lions kicker Jason Hanson, the Lions had Suh attempt an extra point on November 7, 2010, against the New York Jets. However, Suh missed the extra point when the ball hit the right upright.


title: Ndamukong Suh

text: For the 2010 season, Suh led the Lions, all rookies, and all defensive tackles in sacks with 10. He was picked as a starter for the Pro Bowl, becoming the first Lions rookie since Barry Sanders to be picked as a Pro Bowl starter. He missed the game, however, due to shoulder surgery. On January 25, 2011, Suh was named to the All-Pro Team. He was also named the Sporting News Rookie of the Year, the Pro Football Weekly and Pro Football Writers of America Rookie of the Year, the Pepsi NFL Rookie of the Year, and the AP Defensive Rookie of the Year.


title: Ndamukong Suh

text: His rookie year saw the beginning of what would take Suh to a league record amount of fines for on-the-field behavior. He was fined $7,500 in a preseason game against the Cleveland Browns for a facemask of Jake Delhomme, fined $5,000 for using an opponent as leverage on a field goal in a week 9 game against the Jets, and fined $15,000 in a week 13 game against the Bears for unnecessary roughness against quarterback Jay Cutler.


title: Ndamukong Suh

text: During the third quarter of the Lions' Thanksgiving game against the Green Bay Packers on November 24, 2011, Suh pushed Packers' offensive lineman Evan Dietrich-Smith's head into the ground three times, then stomped on his arm. All of this took place after the whistle was blown. Suh was penalized for unnecessary roughness and ejected from the game. The resulting penalty gave the Packers an automatic first down. The Packers would score a touchdown two plays later, giving them a 14–0 lead and an eventual 27–15 victory.


title: Ndamukong Suh

text: Initially, Suh denied stomping on Dietrich-Smith, saying he was only trying to get his balance back. However, on Friday morning, the Lions issued a statement calling Suh's actions "unacceptable". Within hours, Suh wrote on his Facebook page that he'd "made a mistake" a day before and intended to learn from it. Fox Sports NFL rules analyst and former vice president of officiating Mike Pereira said that based on his knowledge of league discipline, "the question won’t be if the NFL will suspend Suh, but when—and for how many games". He drew parallels between Suh's actions and those of Albert Haynesworth, who drew a five-game suspension—the longest suspension for an on-field incident in modern NFL history—for stomping on Andre Gurode's head in 2006.


title: Ndamukong Suh

text: On November 29, the NFL suspended Suh for two games without pay which was $165,294 in lost wages. In announcing the decision, Roger Goodell noted that it was the fifth time Suh had been disciplined for on-field conduct. Suh appealed the decision, and the NFL held an expedited hearing before former Oakland Raiders coach Art Shell, so that a decision could be handed down before the Lions' next game, on December 4, against the New Orleans Saints. The appeal was turned down on December 2, forcing Suh to sit out the game against the Saints and the December 11 game against the Minnesota Vikings.


title: Ndamukong Suh

text: Suh was named a Pro Bowl alternate for the NFC after the 2011 NFL season. He finished the 2011 season with four sacks, one pass defended, and 26 tackles.


title: Ndamukong Suh

text: During a Thanksgiving game on November 22, 2012, Suh was involved in a play in which he kicked Houston Texans quarterback Matt Schaub in the groin. The incident resulted in a $30,000 fine from the league, nearly double the mandated fine for a second offense of "striking/kicking/kneeing" an opponent. There was speculation the size of the fine was recognition by the league of Suh's history of questionable on-field hits, while others viewed it as a "wishy-washy" and "cop-out" action by the league in issuing a large fine without a suspension. In response, NFL Commissioner Roger Goodell said that "intent is something that’s very difficult for us to ever try to make a judgment on". Suh denied the kick was intentional, stating his foot inadvertently hit Schaub as he was being dragged to the ground. Suh finished the 2012 season with eight sacks, two passes defended, and 25 tackles in 16 games (15 starts).


title: Ndamukong Suh

text: In the Lions' Week 1 victory over the Minnesota Vikings, 34–24, Suh low blocked John Sullivan during a DeAndre Levy "pick 6"; the resulting personal foul nullified Levy's touchdown. Two days after the game, Suh was fined $100,000 for the hit—which, not counting lost pay for suspensions, is the largest fine ever issued to a player for on-the-field actions. He appealed the fine but the ruling was later upheld by the NFL.


title: Ndamukong Suh

text: In the Lions' Week 6 victory over the Cleveland Browns, Suh tackled Browns QB Brandon Weeden in a questionable manner. Though not called for a penalty during the game, Suh was fined $31,500 by the NFL, but the fine was later rescinded.


title: Ndamukong Suh

text: During a Thanksgiving game against the Packers on November 28, 2013, Suh sacked Matt Flynn in the end zone, forcing a safety for the first time of his career. On November 29, 2013, Suh was fined $7,875 for performing a throat slash gesture during a game against the Tampa Bay Buccaneers.


title: Ndamukong Suh

text: Suh racked up 53 tackles and 8.5 sacks in the 2014 season. In Week 17, against the Green Bay Packers, Suh stepped on quarterback Aaron Rodgers' calf. Suh received a one-game suspension, seemingly barring him from the Wild Card playoff game the following week against the Dallas Cowboys; but on appeal, arbitrator Ted Cottrell reversed the suspension, opting instead for another fine worth $70,000. Suh played in the Wild Card matchup against the Cowboys and sacked Tony Romo twice.


title: Ndamukong Suh

text: On March 11, 2015, the Miami Dolphins announced that they had signed Suh to a six-year, $114 million contract with $60 million guaranteed. The contract made him the highest-paid defensive player in NFL history, passing Houston Texans defensive end J. J. Watt.


title: Ndamukong Suh

text: During a game against the Washington Redskins on September 13, 2015, as Redskins' running back Alfred Morris was still lying on the ground following a tackle, Suh appeared to knock Morris's helmet off with his leg. The next day, the NFL announced that they would not discipline him, as "Suh's action was not deemed a kick". Suh started all 16 games in 2015, finishing the season with 61 tackles, six sacks, and five passes defended. He was ranked 40th on the NFL Top 100 Players of 2016.


title: Ndamukong Suh

text: In 2016, Suh started all 16 games with 72 tackles, five sacks, and six passes defended. He was ranked 55th by his peers on the NFL Top 100 Players of 2017.


title: Ndamukong Suh

text: During "Thursday Night Football" against the Baltimore Ravens in Week 8, Suh committed two unnecessary roughness penalties, including one moment where he attempted to choke Ryan Mallett and shove him out of reach. Suh claimed that his choke on Mallett was a self-defense, thinking that Mallett tried to attack him first. The Dolphins lost 40–0. Suh finished the season with 48 combined tackles, 4.5 sacks, and two forced fumbles. He was ranked 61st by his peers on the NFL Top 100 Players of 2018.


title: Ndamukong Suh

text: On March 14, 2018, Suh was released, to free up a large amount of salary cap, after playing three seasons with the Dolphins.


title: Ndamukong Suh

text: On March 26, 2018, Suh signed a one-year, $14 million contract with the Los Angeles Rams. On December 2, Suh was fined $20,054 for a horse-collar tackle penalty while facing his former team, the Lions. Suh finished the season with 59 tackles and 4.5 sacks. The Rams finished the season with 13 wins and earned the second seed in the NFC. In the Divisional Round against the Dallas Cowboys, Suh recorded four tackles. In the NFC Championship Game against the New Orleans Saints, Suh had four tackles and 1.5 sacks as the Rams posted a 26–23 overtime victory and Suh reached the Super Bowl for the first time in his career. In the Super Bowl, Suh recorded two tackles in the Rams 13–3 loss to the New England Patriots.


title: Ndamukong Suh

text: The Rams did not offer Suh a new contract and signaled early in the free agent period of the new NFL year that they had decided to move on from him.


title: Ndamukong Suh

text: Suh signed a one-year contract with the Tampa Bay Buccaneers on May 23, 2019. In Week 4, during a 55–40 win over the Los Angeles Rams, Suh recovered a fumble caused by a strip sack by teammate Shaquil Barrett on Jared Goff and returned it 37 yards for a touchdown, the second of his career. In Week 12, during a 35–22 win over the Atlanta Falcons, Suh scored the third fumble return touchdown of his career off a strip sack caused by teammate Jason Pierre-Paul on Matt Ryan. In the process Suh tied a Buccaneers franchise record for the most fumbles returned for touchdowns in a season held by Ronde Barber with his second. In Week 13, during a 28–11 win over the Jacksonville Jaguars, Suh had two tackles and recovered a forced fumble by teammate Carl Nassib. Overall, Suh finished the 2019 season with 41 tackles, 2.5 sacks, four passes defended, four fumble recoveries, and two defensive touchdowns.


title: Ndamukong Suh

text: In Week 2 against the Carolina Panthers, Suh recorded his first two sacks of the season on Teddy Bridgewater during the 31–17 win. Overall, Suh finished the 2020 season with 43 tackles, six sacks, two passes defended, and one forced fumble.


title: Ndamukong Suh

text: In February 2021, Suh won his first Super Bowl as Tampa Bay defeated the Kansas City Chiefs by a score of 31–9 in Super Bowl LV. During the game, Suh recorded 1.5 sacks on Patrick Mahomes.


title: Ndamukong Suh

text: On April 5, 2021, Suh re-signed to a one-year, $9 million contract with the Buccaneers after testing free agency. Suh appeared in all 17 games for the Buccaneers and recorded 27 tackles, 8 tackle for losses, and 6 sacks.


title: Ndamukong Suh

text: Early in his career, Suh was heavily criticized in the media by other players and by the NFL for his fiercely aggressive style of play. Before his second pro season was finished, he had been flagged for nine personal fouls, the most of any player in the league during that time frame. In the first four years of his career, Suh was fined $216,875 by the league for four separate on-field violations. In a poll of fellow players conducted by the "Sporting News", Suh was named "the dirtiest player" in the NFL. He was named the NFL's "Least-Liked Player" in a "Forbes"-publicized Nielsen report in October 2012. While Suh had drawn significant criticism for penalties earlier in his career, his reputation softened towards the end. He was flagged just five times during the entire 2019 season.


title: Ndamukong Suh

text: Suh's mother, Bernadette (née Lennon) Suh, an elementary school teacher, was born in Spanish Town, Jamaica, and is a graduate of Southern Oregon University. His father, Michael Suh, a mechanical engineer, is from Cameroon and played semi-professional soccer in Germany, while also playing for the Cameroonian national team and working as a machinist. They met and married in Portland, Oregon, in 1982, after Michael Suh was admitted to a Portland trade school. Although his father is only , Suh's great-grandfather, also named Ndamukong Suh, stood . In the Ngemba language of Cameroon, Ndamukong means "House of Spears".


title: Ndamukong Suh

text: Suh has a Construction Management degree from the University of Nebraska and has aspirations to be a general contractor after his professional career. His father, Michael Suh, owns his own heating and cooling company in Portland. For his first two years in college before he knew about his professional prospects, Suh had wanted to "work with [his] dad and build his company up to be as big as possible" after graduation.


title: Ndamukong Suh

text: Suh has four sisters; he is the second oldest of the children. His older sister and manager, Odette Lennon Ngum Suh, played soccer collegiately at Mississippi State University and was previously a midfielder on the Cameroon women's national football team. His cousin, Kameron Chatman, played for the Michigan Wolverines men's basketball team.


title: Ndamukong Suh

text: In 2012, Suh participated in Fox's dating game show "The Choice". Suh was nominated on "Splash" on March 10, 2013. He was eliminated on the 2nd week of the show.


title: Ndamukong Suh

text: During the 2013 Stanley Cup Playoffs, Suh rode the Zamboni at Joe Louis Arena wearing a Detroit Red Wings jersey during the Red Wings' playoff series against the Chicago Blackhawks.


title: Ndamukong Suh

text: On April 17, 2010, at the annual Husker Spring Game, Suh announced a $2.6 million donation to the University of Nebraska. Two million dollars of his gift will go to Nebraska Athletics for its Strength and Conditioning Program, and the remaining $600,000 will create an endowed scholarship for the UNL College of Engineering, from which he graduated in 2009 with a degree in construction management. His gift is the largest single charitable contribution by any former player and occurred before Suh was taken with the second overall pick in the 2010 NFL Draft. Suh, a former Grant High School football star, donated $250,000 toward the effort to bring a turf field to Grant High School in 2013.


title: Ndamukong Suh

text: Before Suh was drafted by an NFL team, he signed an endorsement deal with Nike. Suh also signed endorsement deals with Subway, Dick's Sporting Goods, Omaha Steaks, and Chrysler.


title: Megacity

text: A megacity is a very large city, typically with a population of more than 10 million people. Precise definitions vary: the United Nations Department of Economic and Social Affairs in its 2018 "World Urbanization Prospects" report counted urban agglomerations having over 10 million inhabitants. A University of Bonn report held that they are "usually defined as metropolitan areas with a total population of 10 million or more people". Others list cities satisfying criteria of either 5 or 8 million and also having a population density of 2,000 per square kilometre. The terms conurbation, metropolis, and metroplex are also applied to the latter.


title: Megacity

text: The total number of megacities in the world varies between different sources: The world had 33 according to the UN (in 2018), 37 according to "CityPopulation.de" (in 2020), and 35 according to "Demographia" (in 2020). About half these urban agglomerations are in China and India. The other four countries with more than one megacity are Brazil, Japan, Pakistan, and the United States. African megacities are present in Nigeria, Egypt, South Africa, and the DRC; European megacities are present in Russia, France, the United Kingdom, and Turkey (also in Asia); megacities can be found in Latin America in the countries of Brazil, Mexico, Colombia, Peru, and Argentina. Some sources identify Tokyo's Greater Tokyo Area as the largest megacity in the world, while some others give the title to Pearl River Delta.


title: Megacity

text: The term "megacity" entered common use in the late 19th or early 20th centuries; one of the earliest documented uses of the term was by the University of Texas in 1904. Initially the United Nations used the term to describe cities of 8 million or more inhabitants, but now uses the threshold of 10 million. In the mid 1970s the term was coined by urbanist Janise Perlman referring to the phenomenon of very large urban agglomerations.