Manifold Atlas:Definition of “manifold”
This page defines the term “manifold” as used in the Manifold Atlas. We assume that all manifolds are of a fixed dimension n.
- The interior of , denoted , is the subset of points for which is an open subset of .
- The boundary of , written , is the complement of .
- is called closed if is compact and is empty.
A manifold as above is often called a topological manifold for emphasis or clarity. Typically, but not necessarly, the word “manifold” will mean "topological manifold with extra structure", be it piecewise-linear, smooth, complex, symplectic, contact, Riemannian, etc. The extra structure will be emphasised or suppressed in notation and vocabulary as is appropriate. We briefly review some common categories of manifolds below.
2 Atlases of charts
We give a unified presentation of the definition of piecewise linear, smooth and complex manifolds . In the complex case, we assume that the dimension of is even and that the boundary of is empty.
An atlas for is a collection of charts such that the cover .
Let denote either the piecewise linear, smooth or complex categories where by “smooth" we indicate maps. An atlas is a Atlas if every transition function defined by the that atlas is a function: that is, we require every to be either piecewise linear, smooth of class or holomorphic. Two atlases are compatible if their union again forms a atlas and by Zorn's Lemma each atlas defines a unique maximal atlas.
Definition 2.1. A -manifold is a manifold together with a maximal atlas .A -isomorphism is a homeomorphism which is a morphism when viewed in every pair of charts in and .
3 Riemannian Manifolds
A Riemannian metric on a smooth manifold is a smooth family of scalar products
Definition 3.1. A Riemannian manifold is a smooth manifold together with a Riemannian metric .
An isometry between Riemannian manifolds is a diffeomorphism whose differential preserves the metric .