Manifold Atlas:Definition of “manifold”
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== Atlases of charts == | == Atlases of charts == | ||
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− | We give a unified presentation of piecewise linear, smooth and complex manifolds. Recall that a chart on a topological manifold $M$ is a homeomporphism $\phi_\alpha : U_\alpha \to V_\alpha$ 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 | + | We give a unified presentation of piecewise linear, smooth and complex manifolds. In the complex case, we assume that the dimension is even and that the dimension is even. |
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+ | Recall that a chart on a topological manifold $M$ is a homeomporphism $\phi_\alpha : U_\alpha \to V_\alpha$ 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 | ||
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$$ \phi_{\alpha, \beta} : \phi_\alpha(U_\alpha \cap U_\beta) \longrightarrow \phi_\beta(U_\alpha \cap U_\beta).$$ | $$ \phi_{\alpha, \beta} : \phi_\alpha(U_\alpha \cap U_\beta) \longrightarrow \phi_\beta(U_\alpha \cap U_\beta).$$ | ||
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An atlas for $M$ is a collection of charts $A = \{ (U_\alpha, \phi_\alpha)\}$ such that the $U_\alpha$ cover $M$. | 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. That is we require | + | Let $\Cat$ denote either the piecewise linear, smooth or complex categories where by “smooth" we indicate $C^\infty$ maps. 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]]. An atlas is a $\Cat$ Atlas if every transition function defined by the that atlas is a $\Cat$ function. $\Cat$ 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 $A$. | {{beginthm|Definition}} A $\Cat$-manifold $(M, A)$ is a manifold $M$ together with a maximal $\Cat$ atlas $A$. | ||
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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 $B$. | 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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− | + | == Riemannian Manifolds == | |
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− | + | A 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 tangent spaces $T_xM$ for each $x$ in $M$. This means that for each pairof smooth 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 Riemannian manifold $(M, g)$ is a smooth manifold $M$ together with a Riemannian metric $g$. | |
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− | + | An isometry between Riemannian manifolds is a diffeomorphism whose differential preserves the metric $g$. | |
− | + | {{endthm}} | |
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== References == | == References == | ||
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Revision as of 12:30, 18 September 2009
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 the interior 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 piecewise linear, smooth and complex manifolds. In the complex case, we assume that the dimension is even and that the dimension is even.
Recall that a chart on a topological manifold is a homeomporphism 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. That is we require every to be either piecewise linear, smooth of class or holomorphic. An atlas is a Atlas if every transition function defined by the that atlas is a function. 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 .
A -isomorphism is a homeomorphism which is a morphism when viewed in every pair of charts in and .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 pairof 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 .