Maps of manifolds
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
Types of maps
Just as there are various types of manifolds, there are various types of maps of manifolds.
In
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
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
Scalar-valued functions
A basic example of maps between manifolds are scalar-valued functions on a manifold, or sometimes called
In geometric topology, most commonly studied are
In
The
Curves and paths
Dual to scalar-valued functions – maps – are maps which correspond to curves or paths in a manifold. One can also define these where the domain is an interval especially the unit interval or where the domain is a circle (equivalently, a periodic path) S1, which yields a loop. These are used to define the
Embedded paths and loops lead to