Differentiable manifold: Difference between revisions
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===Symbol-free definition=== | ===Symbol-free definition=== | ||
A [[manifold]] is said to be '''differentiable''' if it can be given the structure of a [[differential manifold]], viz if it can be given a compatible differential structure. | A [[manifold]] is said to be '''differentiable''' if it can be given the structure of a [[differential manifold]], viz if it can be given a compatible differential structure. The term '''smooth''' is also sometimes used for this, though the term '''smooth''' might also be used for a differential manifold (i.e., for the manifold along with the differential structure). | ||
==Relation with other properties== | ==Relation with other properties== | ||
===Weaker properties=== | === Weaker properties === | ||
{| class="sortable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
|- | |||
| [[Stronger than::manifold admitting a PL structure]] || manifold that admits the structure of a [[PL manifold]] || (any differential manifold has a unique PL structure compatible with its differential structure) || [[PL not implies differentiable]] || {{intermediate notions short|manifold admitting a PL structure|differentiable manifold}} | |||
|- | |||
| [[Stronger than::triangulable manifold]] || manifold that admits a triangulation, i.e., it is homeomorphic to the geometric realization of a [[simplicial complex]] || (via PL structure, since a PL structure is a particular kind of triangulation) || (via PL) || {{intermediate notions short|triangulable manifold|differentiable manifold}} | |||
|- | |||
| [[Stronger than::polyhedron]] || topological space that admits a triangulation || (via PL structure) || (via PL structure) || {{intermediate notions short|polyhedron|differentiable manifold}} | |||
|- | |||
| [[Stronger than::manifold]] || a topological manifold; no structure beyond that || (by definition) || the [[E8 manifold]] is a counterexample in 4 dimensions || {{intermediate notions short|manifold|differentiable manifold}} | |||
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Latest revision as of 18:31, 22 June 2016
This article defines a property of manifolds and hence also of topological spaces
Definition
Symbol-free definition
A manifold is said to be differentiable if it can be given the structure of a differential manifold, viz if it can be given a compatible differential structure. The term smooth is also sometimes used for this, though the term smooth might also be used for a differential manifold (i.e., for the manifold along with the differential structure).
Relation with other properties
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| manifold admitting a PL structure | manifold that admits the structure of a PL manifold | (any differential manifold has a unique PL structure compatible with its differential structure) | PL not implies differentiable | |FULL LIST, MORE INFO |
| triangulable manifold | manifold that admits a triangulation, i.e., it is homeomorphic to the geometric realization of a simplicial complex | (via PL structure, since a PL structure is a particular kind of triangulation) | (via PL) | |FULL LIST, MORE INFO |
| polyhedron | topological space that admits a triangulation | (via PL structure) | (via PL structure) | Triangulable manifold|FULL LIST, MORE INFO |
| manifold | a topological manifold; no structure beyond that | (by definition) | the E8 manifold is a counterexample in 4 dimensions | Triangulable manifold|FULL LIST, MORE INFO |