Diffeomorphism - Revision history

DavidB4: new link

2020-09-08T21:10:31Z

new link

DavidB4-bot: /* top */clean up & uniformity

2016-07-13T11:40:38Z

top: clean up & uniformity

JacobB at 22:19, December 14, 2009

2009-12-14T22:19:17Z

JacobB: /* Distinct from Homeomorphism */

2009-10-01T21:57:11Z

Distinct from Homeomorphism

MarkGall: 3-manifolds

2009-10-01T21:52:26Z

3-manifolds

MarkGall: historical context

2009-10-01T21:52:11Z

historical context

MarkGall: notational clarification

2009-10-01T04:47:57Z

notational clarification

MarkGall: latex fix

2009-10-01T04:46:55Z

latex fix

MarkGall: examples; a note on how it is different from homeomorphism

2009-10-01T04:45:57Z

examples; a note on how it is different from homeomorphism

Aschlafly: put another way ...

2009-10-01T04:09:48Z

put another way ...

← Older revision		Revision as of 21:10, September 8, 2020
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−	A '''diffeomorphism''' is an infinitely [[~~derivative~~\|differentiable]] [[homeomorphism]] with an infinitely differentiable inverse. Diffeomorphisms provide the right notion for two smooth [[manifolds]] to be "the same": if there exists a diffeomorphism, between two manifolds, then they will have all the same [[topology\|topological]] properties.	+	A '''diffeomorphism''' is an infinitely [[Derivative (calculus)\|differentiable]] [[homeomorphism]] with an infinitely differentiable inverse. Diffeomorphisms provide the right notion for two smooth [[manifolds]] to be "the same": if there exists a diffeomorphism, between two manifolds, then they will have all the same [[topology\|topological]] properties.

	Put another way, a diffeomorphism is a type of function that satisfies the following condition: both the function and its inverse have continuous mixed partial derivatives of all orders in neighborhoods of every possible point.		Put another way, a diffeomorphism is a type of function that satisfies the following condition: both the function and its inverse have continuous mixed partial derivatives of all orders in neighborhoods of every possible point.

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 Put another way, a diffeomorphism is a type of function that satisfies the following condition: both the function and its inverse have continuous mixed partial derivatives of all orders in neighborhoods of every possible point.
-[[category: mathematics]]
+[[Category:Mathematics]]
 == Examples ==

← Older revision		Revision as of 22:19, December 14, 2009
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−	~~{{stub}}~~
	A '''diffeomorphism''' is an infinitely [[derivative\|differentiable]] [[homeomorphism]] with an infinitely differentiable inverse. Diffeomorphisms provide the right notion for two smooth [[manifolds]] to be "the same": if there exists a diffeomorphism, between two manifolds, then they will have all the same [[topology\|topological]] properties.		A '''diffeomorphism''' is an infinitely [[derivative\|differentiable]] [[homeomorphism]] with an infinitely differentiable inverse. Diffeomorphisms provide the right notion for two smooth [[manifolds]] to be "the same": if there exists a diffeomorphism, between two manifolds, then they will have all the same [[topology\|topological]] properties.

← Older revision		Revision as of 21:57, October 1, 2009
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	Two topological spaces are said to be homeomorphic if there exists a homeomorphism between them. In this case, the two spaces are topologically the same. Similarly, two smooth manifolds (which carry the structure of topological spaces) are said to be diffeomorphic if there exists a diffeomorphism between them. Since a diffeomorphism is a special kind of homeomorphism, any two smooth manifolds which are diffeomorphic are in fact homeomorphic. One might expect that if two smooth manifolds are homeomorphic (i.e., equivalent as topological spaces) then they are diffeomorphic (i.e., equivalent as manifolds). However, this is not the case! It is possible to give the 7-dimensional sphere <math>S^7</math> the structure of a smooth manifold in such a way that it is not diffeomorphic to <math>S^7</math> with the standard smooth structure, though it is homeomorphic! In fact, there are exactly 28 such exotic structures on the 7-sphere up to diffeomorphism. Such examples are called '''exotic spheres'''. While the exotic spheres were the first known example of this phenomenon, it was later shown that even 4-dimension Euclidean space may be endowed with any of an uncountable number of non-diffeomorphic smooth structures.		Two topological spaces are said to be homeomorphic if there exists a homeomorphism between them. In this case, the two spaces are topologically the same. Similarly, two smooth manifolds (which carry the structure of topological spaces) are said to be diffeomorphic if there exists a diffeomorphism between them. Since a diffeomorphism is a special kind of homeomorphism, any two smooth manifolds which are diffeomorphic are in fact homeomorphic. One might expect that if two smooth manifolds are homeomorphic (i.e., equivalent as topological spaces) then they are diffeomorphic (i.e., equivalent as manifolds). However, this is not the case! It is possible to give the 7-dimensional sphere <math>S^7</math> the structure of a smooth manifold in such a way that it is not diffeomorphic to <math>S^7</math> with the standard smooth structure, though it is homeomorphic! In fact, there are exactly 28 such exotic structures on the 7-sphere up to diffeomorphism. Such examples are called '''exotic spheres'''. While the exotic spheres were the first known example of this phenomenon, it was later shown that even 4-dimension Euclidean space may be endowed with any of an uncountable number of non-diffeomorphic smooth structures.

−	In some important cases, however, the notions of diffeomorphism and homeomorphism coincide. It is a fundamental theorem of low-dimensional topology that two smooth 3-manifolds are diffeomorphic if and only if they are homeomorphic.	+	In some important cases, however, the notions of diffeomorphism and homeomorphism coincide. It is a fundamental theorem of low-dimensional topology that two smooth 3-manifolds are diffeomorphic if and only if they are homeomorphic. Naturally, this must apply to lower dimensional manifolds as well. This is in stark contrast to higher dimensional manifolds - many 4-manifolds admit an infinite number of smooth structures, and many cannot have any smooth structure at all. Thus, there exist infinite families of 4-manifolds which are all homeomorphic, but none of which are diffeomorphic.

← Older revision		Revision as of 21:52, October 1, 2009
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	== Distinct from Homeomorphism ==		== Distinct from Homeomorphism ==
	Two topological spaces are said to be homeomorphic if there exists a homeomorphism between them. In this case, the two spaces are topologically the same. Similarly, two smooth manifolds (which carry the structure of topological spaces) are said to be diffeomorphic if there exists a diffeomorphism between them. Since a diffeomorphism is a special kind of homeomorphism, any two smooth manifolds which are diffeomorphic are in fact homeomorphic. One might expect that if two smooth manifolds are homeomorphic (i.e., equivalent as topological spaces) then they are diffeomorphic (i.e., equivalent as manifolds). However, this is not the case! It is possible to give the 7-dimensional sphere <math>S^7</math> the structure of a smooth manifold in such a way that it is not diffeomorphic to <math>S^7</math> with the standard smooth structure, though it is homeomorphic! In fact, there are exactly 28 such exotic structures on the 7-sphere up to diffeomorphism. Such examples are called '''exotic spheres'''. While the exotic spheres were the first known example of this phenomenon, it was later shown that even 4-dimension Euclidean space may be endowed with any of an uncountable number of non-diffeomorphic smooth structures.		Two topological spaces are said to be homeomorphic if there exists a homeomorphism between them. In this case, the two spaces are topologically the same. Similarly, two smooth manifolds (which carry the structure of topological spaces) are said to be diffeomorphic if there exists a diffeomorphism between them. Since a diffeomorphism is a special kind of homeomorphism, any two smooth manifolds which are diffeomorphic are in fact homeomorphic. One might expect that if two smooth manifolds are homeomorphic (i.e., equivalent as topological spaces) then they are diffeomorphic (i.e., equivalent as manifolds). However, this is not the case! It is possible to give the 7-dimensional sphere <math>S^7</math> the structure of a smooth manifold in such a way that it is not diffeomorphic to <math>S^7</math> with the standard smooth structure, though it is homeomorphic! In fact, there are exactly 28 such exotic structures on the 7-sphere up to diffeomorphism. Such examples are called '''exotic spheres'''. While the exotic spheres were the first known example of this phenomenon, it was later shown that even 4-dimension Euclidean space may be endowed with any of an uncountable number of non-diffeomorphic smooth structures.
		+
		+	In some important cases, however, the notions of diffeomorphism and homeomorphism coincide. It is a fundamental theorem of low-dimensional topology that two smooth 3-manifolds are diffeomorphic if and only if they are homeomorphic.

← Older revision		Revision as of 21:52, October 1, 2009
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	== Distinct from Homeomorphism ==		== Distinct from Homeomorphism ==
−	Two topological spaces are said to be homeomorphic if there exists a homeomorphism between them. In this case, the two spaces are topologically the same. Similarly, two smooth manifolds (which carry the structure of topological spaces) are said to be diffeomorphic if there exists a diffeomorphism between them. Since a diffeomorphism is a special kind of homeomorphism, any two smooth manifolds which are diffeomorphic are in fact homeomorphic. One might expect that if two smooth manifolds are homeomorphic (i.e., equivalent as topological spaces) then they are diffeomorphic (i.e., equivalent as manifolds). However, this is not the case! It is possible to give the 7-dimensional sphere <math>S^7</math> the structure of a smooth manifold in such a way that it is not diffeomorphic to <math>S^7</math> with the standard smooth structure, though it is homeomorphic! Such examples are called '''exotic spheres'''.	+	Two topological spaces are said to be homeomorphic if there exists a homeomorphism between them. In this case, the two spaces are topologically the same. Similarly, two smooth manifolds (which carry the structure of topological spaces) are said to be diffeomorphic if there exists a diffeomorphism between them. Since a diffeomorphism is a special kind of homeomorphism, any two smooth manifolds which are diffeomorphic are in fact homeomorphic. One might expect that if two smooth manifolds are homeomorphic (i.e., equivalent as topological spaces) then they are diffeomorphic (i.e., equivalent as manifolds). However, this is not the case! It is possible to give the 7-dimensional sphere <math>S^7</math> the structure of a smooth manifold in such a way that it is not diffeomorphic to <math>S^7</math> with the standard smooth structure, though it is homeomorphic! In fact, there are exactly 28 such exotic structures on the 7-sphere up to diffeomorphism. Such examples are called '''exotic spheres'''. While the exotic spheres were the first known example of this phenomenon, it was later shown that even 4-dimension Euclidean space may be endowed with any of an uncountable number of non-diffeomorphic smooth structures.

← Older revision		Revision as of 04:47, October 1, 2009
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	== Distinct from Homeomorphism ==		== Distinct from Homeomorphism ==
−	Two topological spaces are said to be homeomorphic if there exists a homeomorphism between them. In this case, the two spaces are topologically the same. Similarly, two smooth manifolds (which carry the structure of topological spaces) are said to be diffeomorphic if there exists a diffeomorphism between them. Since a diffeomorphism is a special kind of homeomorphism, any two smooth manifolds which are diffeomorphic are in fact homeomorphic. One might expect that if two smooth manifolds are homeomorphic (i.e., equivalent as topological spaces) then they are diffeomorphic (i.e., equivalent as manifolds). However, this is not the case! It is possible to give the sphere <math>S^7</math> the structure of a smooth manifold in such a way that it is not diffeomorphic to <math>S^7</math> with the standard smooth structure, though it is homeomorphic! Such examples are called '''exotic spheres'''.	+	Two topological spaces are said to be homeomorphic if there exists a homeomorphism between them. In this case, the two spaces are topologically the same. Similarly, two smooth manifolds (which carry the structure of topological spaces) are said to be diffeomorphic if there exists a diffeomorphism between them. Since a diffeomorphism is a special kind of homeomorphism, any two smooth manifolds which are diffeomorphic are in fact homeomorphic. One might expect that if two smooth manifolds are homeomorphic (i.e., equivalent as topological spaces) then they are diffeomorphic (i.e., equivalent as manifolds). However, this is not the case! It is possible to give the 7-dimensional sphere <math>S^7</math> the structure of a smooth manifold in such a way that it is not diffeomorphic to <math>S^7</math> with the standard smooth structure, though it is homeomorphic! Such examples are called '''exotic spheres'''.

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 Let <math>\mathbb R^+</math> denote the positive real numbers.  The map <math>\phi : \R \to \R^+</math> defined by <math>\phi(x)=e^x</math> is a diffeomorphism.
-The quotient space <math>\mathbb R/\mathbb Z</math> (that is, the real numbers modulo 1) is diffeomorphic to the unit circle <math>S^1</math> in the complex plane.  <math>\phi : \mathbb R/\mathbb Z \to S^1</math> defined by <math>[x] \mapsto e^{2\pi i x}</math> provides a diffeomorphism.  Note that this map is well-defined: if a and b represent the same class in <math>\mathbb R/\mathbb Z</math>, then <math>a=b+n</math> for some integer n.  Then <math>\phi(a) = e^{2 \pi ia} = e^{2 \pi ia+2\pi n i} = e^{2 \pi i b = \phi(b)</math>.
+The quotient space <math>\mathbb R/\mathbb Z</math> (that is, the real numbers modulo 1) is diffeomorphic to the unit circle <math>S^1</math> in the complex plane.  <math>\phi : \mathbb R/\mathbb Z \to S^1</math> defined by <math>[x] \mapsto e^{2\pi i x}</math> provides a diffeomorphism.  Note that this map is well-defined: if a and b represent the same class in <math>\mathbb R/\mathbb Z</math>, then <math>a=b+n</math> for some integer n.  Then <math>\phi(a) = e^{2 \pi ia} = e^{2 \pi ia+2\pi n i} = e^{2 \pi i b} = \phi(b)</math>.
 == Distinct from Homeomorphism ==
 Two topological spaces are said to be homeomorphic if there exists a homeomorphism between them.  In this case, the two spaces are topologically the same.  Similarly, two smooth manifolds (which carry the structure of topological spaces) are said to be diffeomorphic if there exists a diffeomorphism between them.  Since a diffeomorphism is a special kind of homeomorphism, any two smooth manifolds which are diffeomorphic are in fact homeomorphic.  One might expect that if two smooth manifolds are homeomorphic (i.e., equivalent as topological spaces) then they are diffeomorphic (i.e., equivalent as manifolds).  However, this is not the case!  It is possible to give the sphere <math>S^7</math> the structure of a smooth manifold in such a way that it is not diffeomorphic to <math>S^7</math> with the standard smooth structure, though it is homeomorphic!  Such examples are called '''exotic spheres'''.

@@ Line 1: / Line 1: @@
 {{stub}}
-A '''diffeomorphism''' is an infinitely [[derivative|differentiable]] [[homeomorphism]] with an infinitely differentiable inverse.
+A '''diffeomorphism''' is an infinitely [[derivative|differentiable]] [[homeomorphism]] with an infinitely differentiable inverse.  Diffeomorphisms provide the right notion for two smooth [[manifolds]] to be "the same": if there exists a diffeomorphism, between two manifolds, then they will have all the same [[topology|topological]] properties.
 Put another way, a diffeomorphism is a type of function that satisfies the following condition: both the function and its inverse have continuous mixed partial derivatives of all orders in neighborhoods of every possible point.
 [[category: mathematics]]

← Older revision		Revision as of 04:09, October 1, 2009
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	A '''diffeomorphism''' is an infinitely [[derivative\|differentiable]] [[homeomorphism]] with an infinitely differentiable inverse.		A '''diffeomorphism''' is an infinitely [[derivative\|differentiable]] [[homeomorphism]] with an infinitely differentiable inverse.

		+	Put another way, a diffeomorphism is a type of function that satisfies the following condition: both the function and its inverse have continuous mixed partial derivatives of all orders in neighborhoods of every possible point.
	[[category: mathematics]]		[[category: mathematics]]