Manifold Atlas:Definition of “manifold”

Definition 1.1.

An n-dimensional manifold

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$== Introduction == <wikitex>; This page defines the term “manifold” as used in the Manifold Atlas. <!-- and summarises a few key properties of all manifolds --> 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. </wikitex> == Atlases of charts == <wikitex>; 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|homeomporphism]] $\phi_\alpha : U_\alpha \to V_\alpha$ 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 $B$. </wikitex> == Riemannian Manifolds == <wikitex>; 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}} </wikitex> <!-- == References == {{#RefList:}} --> <!--[[Category:Theory]]--> <!-- {{Stub}} -->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

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  $M$, denoted $\mathop{\mathrm{int}}(M)$, is the subset of points for which $U_x$ is an open subset of $\Rr^n$.
• The boundary of

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  $M$, written $\partial M$, is the complement of $\mathop{\mathrm{int}}(M)$.

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  $M$ is called closed if

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  $M$ is compact and $\partial M$ is empty.

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Definition 2.1. A $\Cat$-manifold $(M, A)$ is a manifold

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$M$ 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 $B$.

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Definition 3.1.

A Riemannian manifold $(M, g)$ is a smooth manifold

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$M$ together with a Riemannian metric $g$.

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