page_title: Maps of manifolds
section_title: Types of maps
hierarchical_section_title: Maps of manifolds / Types of maps
context_page_description: In mathematics, more specifically in differential geometry and topology, various types of functions between manifolds are studied, both as objects in their own right and for the light they shed
context_section_description: Just as there are various types of manifolds, there are various types of maps of manifolds.
  In geometric topology, the basic types of maps correspond to various categories of manifolds: DIFF for smooth functions between differentiable manifolds, PL for piecewise linear functions between piecewise linear manifolds, and TOP for continuous functions between topological manifolds. These are progressively weaker structures, properly connected via PDIFF, the category of piecewise-smooth maps between piecewise-smooth manifolds.
  In addition to these general categories of maps, there are maps with special properties; these may or may not form categories, and may or may not be generally discussed categorically.
  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. Basic results include the Whitney embedding theorem and Whitney immersion theorem.
  In complex geometry, ramified covering spaces are used to model Riemann surfaces, and to analyze maps between surfaces, such as by the Riemann–Hurwitz formula.
  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.
caption: The right-handed trefoil knot.