Manifold Atlas:Definition of “manifold”
1 Introduction
This page defines the term “manifold” as used in the Manifold Atlas. We assume that all manifolds are of a fixed dimension n.
Definition 1.1. An n-dimensional manifold is a second countable, Hausdorff space for which every point has a neighbourhood homeomorphic to an open subset of .
- 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.
Recall that a chart on a topological manifold is a homeomporphism from an open subset of to an open subset of . The transition function defined by two charts and is the homeomorphism
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
defined on the tangent spaces for each in . This means that for each pair of smooth vector fields and on the map
is smooth.
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 .