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 $M$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}M$ is a second countable, Hausdorff space for which every point $x \in M$$x \in M$ has a neighbourhood $U_x$$U_x$ homeomorphic to an open subset of $\Rr^n_+ := \{ v \in \Rr^n | v_1 \geq 0 \}$$\Rr^n_+ := \{ v \in \Rr^n | v_1 \geq 0 \}$.

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

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

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

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

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

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

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

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

3 Riemannian Manifolds

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

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

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

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

is smooth.

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

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