Manifold Atlas:Definition of “manifold”

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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 M is a second countable, Hausdorff space for which every point x \in M has a neighbourhood U_x homeomorphic to an open subset of \Rr^n_+ := \{ v \in \Rr^n | v_1 \geq 0 \}.

  • The interior of M, denoted \mathop{\mathrm{int}}(M), is the subset of points for which U_x is an open subset of \Rr^n.
  • The boundary of M, written \partial M, is the complement of \mathop{\mathrm{int}}(M).
  • M is called closed if M is compact and \partial M is empty.

A manifold M 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 M. In the complex case, we assume that the dimension of M is even and that the boundary of M is empty.

Recall that a chart on a topological manifold M is a homeomorphism \phi_\alpha : U_\alpha \to V_\alpha from an open subset U_\alpha of M to an open subset V_\alpha of \Rr^n_+. The transition function defined by two charts \phi_\alpha and \phi_\beta is the homeomorphism

\displaystyle  \phi_{\alpha\beta} := \phi_\alpha \circ \phi_\beta^{-1}|_{\phi_{\beta}(U_\alpha \cap U_\beta)} : \phi_\beta(U_\alpha \cap U_\beta) \longrightarrow \phi_\alpha(U_\alpha \cap U_\beta).

An atlas for M is a collection of charts A = \{ (U_\alpha, \phi_\alpha)\} such that the U_\alpha cover M.

Let \Cat denote either the piecewise linear, smooth or complex categories where by “smooth" we indicate C^\infty maps. An atlas is a \Cat Atlas if every transition function defined by the that atlas is a \Cat function: that is, we require every \phi_{\alpha, \beta} to be either piecewise linear, smooth of class C^\infty or holomorphic. Two \Cat atlases are compatible if their union again forms a \Cat atlas and by Zorn's Lemma each \Cat atlas defines a unique maximal \Cat atlas.

Definition 2.1. A \Cat-manifold (M, A) is a manifold M together with a maximal \Cat atlas A.

A \Cat-isomorphism (M, A) \cong (N, B) is a homeomorphism f: M \cong N which is a \Cat morphism when viewed in every pair of charts in A and B.

3 Riemannian Manifolds

A Riemannian metric g on a smooth manifold M is a smooth family of scalar products

\displaystyle  g_x : T_xM \times T_xM \longmapsto \Rr

defined on the tangent spaces T_xM for each x in M. This means that for each pair of smooth vector fields v_1 and v_2 on M the map

\displaystyle  M \to \Rr, ~~~ x \longmapsto g_x(v_1(x),v_2(x))

is smooth.

Definition 3.1. A Riemannian manifold (M, g) is a smooth manifold M together with a Riemannian metric g.

An isometry between Riemannian manifolds is a diffeomorphism whose differential preserves the metric g.

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