Manifold Atlas:Definition of “manifold”
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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. | + | 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. |
$Cat$ 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. | $Cat$ 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. | ||
Revision as of 21:08, 16 September 2009
Contents |
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 manifoldTex syntax erroris a second countable Hausdorff space for which every point has a neighbourhood homeomorphic to an open subset of .
- The interior of
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, denoted , is the subset of points for which . - The boundary of
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, written , is the complement of the interior ofTex syntax error
. -
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is called closed ifTex syntax error
is compact and is empty.
Tex syntax erroras 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 smooth, Riemannian, complex, 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
Tex syntax erroris a homeomporphism from an open subset of
Tex syntax errorto an open subset of . The transition function defined by two charts and is the homeomorphism
Tex syntax erroris a collection of charts such that the cover
Tex syntax error.
Let denote either the piecewise linear, smooth or complex categories where by “smooth" we indicate maps. 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.
Tex syntax errortogether 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
4 References
This page has not been refereed. The information given here might be incomplete or provisional. |