# 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$$== Introduction == ; This page defines the term “manifold” as used in the Manifold Atlas. We assume that all manifolds are of a fixed dimension n. {{beginthm|Definition|}} An '''n-dimensional manifold''' M is a [[Wikipedia:Second_countable|second countable]], [[Wikipedia:Hausdorff_space|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. {{endthm}} 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 [[Wikipedia:Piecewise_linear_manifold|piecewise-linear]], [[Wikipedia:Differential_manifold|smooth]], [[Wikipedia:Complex_manifold|complex]], [[Wikipedia:Symplectic_manifold|symplectic]], [[Wikipedia:Contact_manifold|contact]], [[Wikipedia:Riemannian_manifold|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. == 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 [[Wikipedia:Homeomorphism|homeomorphism]] \phi_\alpha : U_\alpha \to V_\alpha from an open subset U_\alpha of M to an [[Wikipedia:Open_set|open subset]] V_\alpha of \Rr^n_+. The transition function defined by two charts \phi_\alpha and \phi_\beta is the homeomorphism \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 [[Wikipedia:Differential_manifold#Atlases|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 [[Wikipedia:Piecewise_linear_function#Notation|piecewise linear]], [[Wikipedia:Smooth_function#Differentiability_classes_in_several_variables|smooth of class C^\infty]] or [[Wikipedia:Holomorphic_function|holomorphic]]. Two \Cat atlases are compatible if their union again forms a \Cat atlas and by [[Wikipedia:Zorn's_lemma|Zorn's Lemma]] each \Cat atlas defines a unique maximal \Cat atlas. {{beginthm|Definition}} 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. == Riemannian Manifolds == ; A [[Wikipedia:Riemannian_manifold#Riemannian metrics|Riemannian metric]] g on a smooth manifold M is a smooth family of scalar products g_x : T_xM \times T_xM \longmapsto \Rr defined on the [[Wikipedia:Tangent_space|tangent spaces]] T_xM for each x in M. This means that for each pair of smooth [[Wikipedia:Vector_field|vector fields]] v_1 and v_2 on M the map M \to \Rr, ~~~ x \longmapsto g_x(v_1(x),v_2(x)) is smooth. {{beginthm|Definition}} A [[Wikipedia:Riemannian_manifold|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. {{endthm}} 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

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

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$$\Cat$ denote either the piecewise linear, smooth or complex categories where by “smooth" we indicate
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$C^\infty$ maps. An atlas is a $\Cat$$\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
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$C^\infty$
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

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

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

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

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$.