
Covariant derivative

In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically embedded into a higher-dimensional Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean directional derivative onto the manifold's tangent space. In this case the Euclidean derivative is broken into two parts, the extrinsic normal component (dependent on the embedding) and the intrinsic covariant derivative component.
The name is motivated by the importance of changes of coordinate in physics: the covariant derivative transforms covariantly under a general coordinate transformation, that is, linearly via the Jacobian matrix of the transformation.
This article presents an introduction to the covariant derivative of a vector field with respect to a vector field, both in a coordinate free language and using a local coordinate system and the traditional index notation. The covariant derivative of a tensor field is presented as an extension of the same concept. The covariant derivative generalizes straightforwardly to a notion of differentiation associated to a connection on a vector bundle, also known as a Koszul connection.
History.
Historically, at the turn of the 20th century, the covariant derivative was introduced by Gregorio Ricci-Curbastro and Tullio Levi-Civita in the theory of Riemannian and pseudo-Riemannian geometry. Ricci and Levi-Civita (following ideas of Elwin Bruno Christoffel) observed that the Christoffel symbols used to define the curvature could also provide a notion of differentiation which generalized the classical directional derivative of vector fields on a manifold. This new derivative – the Levi-Civita connection – was "covariant" in the sense that it satisfied Riemann's requirement that objects in geometry should be independent of their description in a particular coordinate system.
It was soon noted by other mathematicians, prominent among these being Hermann Weyl, Jan Arnoldus Schouten, and Élie Cartan, that a covariant derivative could be defined abstractly without the presence of a metric. The crucial feature was not a particular dependence on the metric, but that the Christoffel symbols satisfied a certain precise second order transformation law. This transformation law could serve as a starting point for defining the derivative in a covariant manner. Thus the theory of covariant differentiation forked off from the strictly Riemannian context to include a wider range of possible geometries.
In the 1940s, practitioners of differential geometry began introducing other notions of covariant differentiation in general vector bundles which were, in contrast to the classical bundles of interest to geometers, not part of the tensor analysis of the manifold. By and large, these generalized covariant derivatives had to be specified "ad hoc" by some version of the connection concept. In 1950, Jean-Louis Koszul unified these new ideas of covariant differentiation in a vector bundle by means of what is known today as a Koszul connection or a connection on a vector bundle. Using ideas from Lie algebra cohomology, Koszul successfully converted many of the analytic features of covariant differentiation into algebraic ones. In particular, Koszul connections eliminated the need for awkward manipulations of Christoffel symbols (and other analogous non-tensorial objects) in differential geometry. Thus they quickly supplanted the classical notion of covariant derivative in many post-1950 treatments of the subject.
Motivation.
The covariant derivative is a generalization of the directional derivative from vector calculus. As with the directional derivative, the covariant derivative is a rule, formula_1, which takes as its inputs: (1) a vector, u, defined at a point "P", and (2) a vector field, v, defined in a neighborhood of "P". The output is the vector formula_2, also at the point "P". The primary difference from the usual directional derivative is that formula_1 must, in a certain precise sense, be "independent" of the manner in which it is expressed in a coordinate system.
A vector may be "described" as a list of numbers in terms of a basis, but as a geometrical object a vector retains its own identity regardless of how one chooses to describe it in a basis. This persistence of identity is reflected in the fact that when a vector is written in one basis, and then the basis is changed, the components of the vector transform according to a change of basis formula. Such a transformation law is known as a covariant transformation. The covariant derivative is required to transform, under a change in coordinates, in the same way as a basis does: the covariant derivative must change by a covariant transformation (hence the name).
In the case of Euclidean space, one tends to define the derivative of a vector field in terms of the difference between two vectors at two nearby points.
In such a system one translates one of the vectors to the origin of the other, keeping it parallel. With a Cartesian (fixed orthonormal) coordinate system "keeping it parallel" amounts to keeping the components constant. Euclidean space provides the simplest example: a covariant derivative which is obtained by taking the ordinary directional derivative of the components in the direction of the displacement vector between the two nearby points.
In the general case, however, one must take into account the change of the coordinate system. For example, if the same covariant derivative is written in polar coordinates in a two dimensional Euclidean plane, then it contains extra terms that describe how the coordinate grid itself "rotates". In other cases the extra terms describe how the coordinate grid expands, contracts, twists, interweaves, etc. In this case "keeping it parallel" does "not" amount to keeping components constant under translation.
Consider the example of moving along a curve "γ"("t") in the Euclidean plane. In polar coordinates, γ may be written in terms of its radial and angular coordinates by "γ"("t") = ("r"("t"), "θ"("t")). A vector at a particular time "t" (for instance, the acceleration of the curve) is expressed in terms of formula_4, where formula_5 and formula_6 are unit tangent vectors for the polar coordinates, serving as a basis to decompose a vector in terms of radial and tangential components. At a slightly later time, the new basis in polar coordinates appears slightly rotated with respect to the first set. The covariant derivative of the basis vectors (the Christoffel symbols) serve to express this change.
In a curved space, such as the surface of the Earth (regarded as a sphere), the translation is not well defined and its analog, parallel transport, depends on the path along which the vector is translated.
A vector e on a globe on the equator at point Q is directed to the north. Suppose we parallel transport the vector first along the equator until at point P and then (keeping it parallel to itself) drag it along a meridian to the pole N and (keeping the direction there) subsequently transport it along another meridian back to Q. Then we notice that the parallel-transported vector along a closed circuit does not return as the same vector; instead, it has another orientation. This would not happen in Euclidean space and is caused by the "curvature" of the surface of the globe. The same effect can be noticed if we drag the vector along an infinitesimally small closed surface subsequently along two directions and then back. The infinitesimal change of the vector is a measure of the curvature.
Informal definition using an embedding into Euclidean space.
Suppose an open subset formula_8 of a Riemannian manifold formula_9 is embedded into Euclidean space formula_10 via a twice continuously-differentiable (C) mapping formula_11 such that the tangent space at formula_12 is spanned by the vectors
and the scalar product formula_14 on formula_15 is compatible with the metric on "M":
For a tangent vector field, one has
The last term is not tangential to "M", but can be expressed as a linear combination of the tangent space base vectors using the Christoffel symbols as linear factors plus a vector orthogonal to the tangent space:
In the case of the Levi-Civita connection, the covariant derivative formula_19, also written is defined as the orthogonal projection of the usual derivative onto tangent space:
To obtain the relation between Christoffel symbols for the Levi-Civita connection and the metric, first we must note that, since formula_21 in previous equation is orthogonal to tangent space:
Second, the partial derivative of a component of the metric is:
implies for a basis using the symmetry of the scalar product and swapping the order of partial differentiations: 
adding first row to second and subtracting third one:
and yields the Christoffel symbols for the Levi-Civita connection in terms of the metric:
For a very simple example that captures the essence of the description above, draw a circle on a flat sheet of paper. Travel around the circle at a constant speed. The derivative of your velocity, your acceleration vector, always points radially inward. Roll this sheet of paper into a cylinder. Now the (Euclidean) derivative of your velocity has a component that sometimes points inward toward the axis of the cylinder depending on whether you're near a solstice or an equinox. (At the point of the circle when you are moving parallel to the axis, there is no inward acceleration. Conversely, at a point (1/4 of a circle later) when the velocity is along the cylinder's bend, the inward acceleration is maximum.) This is the (Euclidean) normal component. The covariant derivative component is the component parallel to the cylinder's surface, and is the same as that before you rolled the sheet into a cylinder.
Formal definition.
A covariant derivative is a (Koszul) connection on the tangent bundle and other tensor bundles: it differentiates vector fields in a way analogous to the usual differential on functions. The definition extends to a differentiation on the duals of vector fields (i.e. covector fields) and to arbitrary tensor fields, in a unique way that ensures compatibility with the tensor product and trace operations (tensor contraction).
Functions.
Given a point formula_27 of the manifold formula_9, a real function formula_29 on the manifold and a tangent vector formula_30, the covariant derivative of "f" at "p" along v is the scalar at "p", denoted formula_31, that represents the principal part of the change in the value of "f" when the argument of "f" is changed by the infinitesimal displacement vector v. (This is the differential of "f" evaluated against the vector v.) Formally, there is a differentiable curve formula_32 such that formula_33 and formula_34, and the covariant derivative of "f" at "p" is defined by
When formula_36 is a vector field on formula_9, the covariant derivative formula_38 is the function that associates with each point "p" in the common domain of "f" and v the scalar formula_39. This coincides with the usual Lie derivative of "f" along the vector field v.
Vector fields.
Given a point formula_40 of the manifold formula_9, a vector field formula_42 defined in a neighborhood of "p" and a tangent vector formula_30, the covariant derivative of u at "p" along v is the tangent vector at "p", denoted formula_44, such that the following properties hold (for any tangent vectors v, x and y at "p", vector fields u and w defined in a neighborhood of "p", scalar values "g" and "h" at "p", and scalar function "f" defined in a neighborhood of "p"):
Note that formula_45 depends not only on the value of u at "p" but also on values of u in an infinitesimal neighbourhood of "p" because of the last property, the product rule.
If u and v are both vector fields defined over a common domain, then formula_55 denotes the vector field whose value at each point "p" of the domain is the tangent vector formula_56. 
Covector fields.
Given a field of covectors (or one-form) formula_57 defined in a neighborhood of "p", its covariant derivative formula_58 is defined in a way to make the resulting operation compatible with tensor contraction and the product rule. That is, formula_58 is defined as the unique one-form at "p" such that the following identity is satisfied for all vector fields u in a neighborhood of "p"
The covariant derivative of a covector field along a vector field v is again a covector field.
Tensor fields.
Once the covariant derivative is defined for fields of vectors and covectors it can be defined for arbitrary tensor fields by imposing the following identities for every pair of tensor fields formula_61 and formula_62 in a neighborhood of the point "p":
and for formula_64 and formula_65 of the same valence
The covariant derivative of a tensor field along a vector field v is again a tensor field of the same type.
Explicitly, let "T" be a tensor field of type . Consider "T" to be a differentiable multilinear map of smooth sections "α"1, "α"2, …, "α""q" of the cotangent bundle "T"∗"M" and of sections "X"1, "X"2, …, "X""p" of the tangent bundle "TM", written "T"("α"1, "α"2, …, "X"1, "X"2, …) into R. The covariant derivative of "T" along "Y" is given by the formula
Coordinate description.
Given coordinate functions
any tangent vector can be described by its components in the basis
The covariant derivative of a basis vector along a basis vector is again a vector and so can be expressed as a linear combination formula_70.
To specify the covariant derivative it is enough to specify the covariant derivative of each basis vector field formula_71 along formula_72.
the coefficients formula_74 are the components of the connection with respect to a system of local coordinates. In the theory of Riemannian and pseudo-Riemannian manifolds, the components of the Levi-Civita connection with respect to a system of local coordinates are called Christoffel symbols.
Then using the rules in the definition, we find that for general vector fields formula_75 and formula_76 we get
so
The first term in this formula is responsible for "twisting" the coordinate system with respect to the covariant derivative and the second for changes of components of the vector field "u". In particular
In words: the covariant derivative is the usual derivative along the coordinates with correction terms which tell how the coordinates change.
For covectors similarly we have
where formula_81.
The covariant derivative of a type tensor field along formula_82 is given by the expression:
Or, in words: take the partial derivative of the tensor and add: formula_84 for every upper index formula_85, and formula_86 for every lower index formula_87.
If instead of a tensor, one is trying to differentiate a "tensor density" (of weight +1), then you also add a term
If it is a tensor density of weight "W", then multiply that term by "W".
For example, formula_89 is a scalar density (of weight +1), so we get:
where semicolon ";" indicates covariant differentiation and comma "," indicates partial differentiation. Incidentally, this particular expression is equal to zero, because the covariant derivative of a function solely of the metric is always zero.
Notation.
In textbooks on physics, the covariant derivative is sometimes simply stated in terms of its components in this equation.
Often a notation is used in which the covariant derivative is given with a semicolon, while a normal partial derivative is indicated by a comma. In this notation we write the same as:
In case two or more indexes after the semicolon, all them must be understood as covariant derivatives:
In some older texts (notably Adler, Bazin & Schiffer, "Introduction to General Relativity"), the covariant derivative is denoted by a double pipe and the partial derivative by single pipe:
Covariant derivative by field type.
For a scalar field formula_94, covariant differentiation is simply partial differentiation:
For a contravariant vector field formula_96, we have:
For a covariant vector field formula_98, we have:
For a type (2,0) tensor field formula_100, we have:
For a type (0,2) tensor field formula_102, we have:
For a type (1,1) tensor field formula_104, we have:
The notation above is meant in the sense
Properties.
In general, covariant derivatives do not commute. By example, the covariant derivatives of vector field formula_107. Riemann tensor formula_108is defined such that:
or, equivalently,
The covariant derivative of a (2,1)-tensor field fulfills: 
The latter can be shown by taking (without loss of generality) that formula_112.
Derivative along a curve.
Since the covariant derivative formula_113 of a tensor field formula_114 at a point formula_40 depends only on the value of the vector field formula_116 at formula_40 one can define the covariant derivative along a smooth curve formula_118 in a manifold:
Note that the tensor field formula_114 only needs to be defined on the curve formula_118 for this definition to make sense.
In particular, formula_122 is a vector field along the curve formula_123 itself. If formula_124 vanishes then the curve is called a geodesic of the covariant derivative. If the covariant derivative is the Levi-Civita connection of a positive-definite metric then the geodesics for the connection are precisely the geodesics of the metric that are parametrised by arc length.
The derivative along a curve is also used to define the parallel transport along the curve.
Sometimes the covariant derivative along a curve is called absolute or intrinsic derivative.
Relation to Lie derivative.
A covariant derivative introduces an extra geometric structure on a manifold that allows vectors in neighboring tangent spaces to be compared: there is no canonical way to compare vectors from different tangent spaces because there is no canonical coordinate system.
There is however another generalization of directional derivatives which "is" canonical: the Lie derivative, which evaluates the change of one vector field along the flow of another vector field. Thus, one must know both vector fields in an open neighborhood, not merely at a single point. The covariant derivative on the other hand introduces its own change for vectors in a given direction, and it only depends on the vector direction at a single point, rather than a vector field in an open neighborhood of a point. In other words, the covariant derivative is linear (over "C"∞("M")) in the direction argument, while the Lie derivative is linear in neither argument.
Note that the antisymmetrized covariant derivative , and the Lie derivative "L""u""v" differ by the torsion of the connection, so that if a connection is torsion free, then its antisymmetrization "is" the Lie derivative.