# 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 homeomorphism 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 .

## 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 .