Differentiable manifold

In mathematics , differentiable manifolds are a generic term for curves , surfaces and other geometric objects which - from the perspective of analysis - look locally like a Euclidean space . In contrast to topological manifolds , on differentiable manifolds it is possible to talk about derivatives and related concepts. Differentiable manifolds are the main subject of differential geometry and differential topology . They also play a central role in theoretical physics, especially in classical mechanics in systems subject to constraints, and in the description of spacetime in general relativity .

There are two approaches to differentiable manifolds: on the one hand, as subsets of a higher-dimensional Euclidean space , which are described either by equations or by parameterizations and are dealt with in the article Submanifolds of the ${\ displaystyle \ mathbb {R} ^ {n}}$ , and on the other hand as abstract manifolds whose differentiable structure is given by an atlas . The equivalence of the two perspectives is ensured by Whitney's embedding theorem .

Definitions

Differentiable atlas

The graphic illustrates a card change of cards and . The large circle symbolizes the topological space and the two lower smaller circles symbolize subsets of the .

{\ displaystyle (U, \ phi)}

{\ displaystyle (V, \ psi)}

{\ displaystyle \ mathbb {R} ^ {n}}

A map of a topological space is a pair consisting of an in open, non-empty set and a homeomorphism ${\ displaystyle M}$ ${\ displaystyle (U, \ phi)}$ ${\ displaystyle M}$ ${\ displaystyle U \ subseteq M}$

{\ displaystyle \ phi \ colon \ U \ to \ phi (U) \ subseteq \ mathbb {R} ^ {n}}

.

If and are two cards with , then the figure is called ${\ displaystyle (U, \ phi)}$ ${\ displaystyle (V, \ psi)}$ ${\ displaystyle M}$ ${\ displaystyle U \ cap V \ neq \ emptyset}$

{\ displaystyle \ psi \ circ \ phi ^ {- 1} \ colon \ \ phi (U \ cap V) \ to \ psi (U \ cap V)}

a card change .

An atlas for is then a family of maps ( is an index set) such that ${\ displaystyle M}$ ${\ displaystyle (U_ {i}, \ phi _ {i}) _ {i \ in I}}$ ${\ displaystyle I}$

{\ displaystyle M = \ bigcup _ {i \ in I} U_ {i}}

applies. An atlas is called -differentiable with if all of its map changes are -diffeomorphisms . ${\ displaystyle C ^ {k}}$ ${\ displaystyle k \ geq 1}$ ${\ displaystyle C ^ {k}}$

Differentiable structure

Two -differentiable atlases are equivalent , even if their union is a -differentiable atlas. An equivalence class of atlases with regard to this equivalence relation is called -differentiable structure of the manifold. ${\ displaystyle C ^ {k}}$ ${\ displaystyle C ^ {k}}$ ${\ displaystyle C ^ {k}}$

If it is, one also speaks of a smooth structure . ${\ displaystyle k = \ infty}$

Differentiable manifold

A -time differentiable manifold is a topological Hausdorff space that satisfies the second axiom of countability , together with a -differentiable structure. ${\ displaystyle k}$ ${\ displaystyle C ^ {k}}$

The differentiable manifold has the dimension when a map and thus all maps map into a subset of the . ${\ displaystyle n}$ ${\ displaystyle \ mathbb {R} ^ {n}}$

Smooth manifold

A smooth manifold is also a topological Hausdorff space that satisfies the second axiom of countability, together with a smooth structure.

On smooth manifolds one can examine functions for smoothness , which is of course not possible with -times differentiable manifolds, since there the map change is only -times differentiable and one can therefore differentiate each function on the manifold only -times. Often, differential geometers only consider the smooth manifolds, since for these one obtains roughly the same results as for those that can be differentiated, but does not have to manage how often one can still differentiate the map changes. ${\ displaystyle k}$ ${\ displaystyle k}$ ${\ displaystyle k}$ ${\ displaystyle k}$

Complex manifold

Complex manifolds are also smooth, but with the addition that the map changes are also biholomorphic .

Examples

The two-dimensional sphere

The Euclidean vector space can also be understood as a -dimensional differentiable manifold. A differentiable atlas consisting of a map is obtained by means of the identical illustration . ${\ displaystyle \ mathbb {R} ^ {n}}$ ${\ displaystyle n}$

Probably the simplest but nontrivial example of a differentiable manifold is the -dimensional sphere . The two-dimensional sphere can be imagined as the envelope of a sphere . A differentiable atlas of the sphere can be obtained with the help of two maps, for example with the help of stereographic projection . On the sphere, however, it is possible, depending on the dimension, to define different incompatible differentiable atlases. ${\ displaystyle n}$

Differentiable images, paths and functions

If a -dimensional and a -dimensional -manifold, then a continuous mapping is called a mapping or times continuously differentiable (short: differentiable ), if this applies to your map representations (these are maps from to ). ${\ displaystyle M}$ ${\ displaystyle m}$ ${\ displaystyle N}$ ${\ displaystyle n}$ ${\ displaystyle C ^ {k}}$ ${\ displaystyle F \ colon M \ to N}$ ${\ displaystyle C ^ {k}}$ ${\ displaystyle k}$ ${\ displaystyle \ mathbb {R} ^ {m}}$ ${\ displaystyle \ mathbb {R} ^ {n}}$

In detail: If a card is from and a card is from with , it is called ${\ displaystyle (U, \ phi)}$ ${\ displaystyle M}$ ${\ displaystyle (V, \ psi)}$ ${\ displaystyle N}$ ${\ displaystyle F (U) \ subset V}$

{\ displaystyle \ psi \ circ F \ circ \ phi ^ {- 1} \ colon \ phi (U) \ to \ psi (V)}

a map representation of (with regard to the two maps). ${\ displaystyle F}$

The mapping is now called continuously differentiable by the class or times if all map representations are of the class . The differentiability does not depend on the choice of cards. This results from the fact that the map change maps are diffeomorphisms and from the multidimensional chain rule . The continuity of does not follow from the differentiability, but must be assumed so that the cards can be chosen in such a way that applies. ${\ displaystyle F}$ ${\ displaystyle C ^ {k}}$ ${\ displaystyle k}$ ${\ displaystyle C ^ {k}}$ ${\ displaystyle C ^ {k}}$ ${\ displaystyle F}$ ${\ displaystyle F (U) \ subset V}$

Maps of the class , which can be differentiated as often as required, are also referred to as smooth maps . ${\ displaystyle C ^ {\ infty}}$

The cases or are also possible. In this case there is no need for the cards. ${\ displaystyle M = \ mathbb {R} ^ {m}}$ ${\ displaystyle N = \ mathbb {R} ^ {n}}$

A differentiable mapping of an interval into a manifold is called a path or parameterized curve . If the target area is , one speaks of a differentiable function . ${\ displaystyle I \ subset \ mathbb {R}}$ ${\ displaystyle \ mathbb {R}}$ ${\ displaystyle M}$

A map is called local diffeomorphism if the maps can be chosen in such a way that the map representations are of diffeomorphisms. If it is also bijective, it is called a -diffeomorphism. ${\ displaystyle F \ colon M \ to N}$ ${\ displaystyle C ^ {k}}$ ${\ displaystyle F}$ ${\ displaystyle F}$ ${\ displaystyle F}$ ${\ displaystyle C ^ {k}}$

In order to actually be able to define a derivative for mappings between differentiable manifolds, one needs an additional structure, the tangent space . For the definition of the derivative of a differentiable mapping between manifolds see tangent space and push forward .

properties

The diffeomorphism group operates transitively on a connected differentiable manifold , that is, there is a diffeomorphism for all , so that holds. ${\ displaystyle M}$ ${\ displaystyle x, y \ in M}$ ${\ displaystyle F \ colon M \ to M}$ ${\ displaystyle F (x) = y}$

The class of -manifolds and the class of -maps form a category . ${\ displaystyle C ^ {k}}$ ${\ displaystyle C ^ {k}}$

Differentiable manifolds are triangulable , which is generally not the case for topological manifolds.

Submanifolds

A -dimensional submanifold of a -dimensional manifold ( ) is a subset that appears in suitable maps like a -dimensional linear subspace of the . This has a differentiable structure in canonical fashion. ${\ displaystyle n}$ ${\ displaystyle m}$ ${\ displaystyle M}$ ${\ displaystyle n <m}$ ${\ displaystyle n}$ ${\ displaystyle \ mathbb {R} ^ {m}}$

In detail: A subset of a -dimensional differentiable manifold is a -dimensional submanifold , if for every point there is a map around such that ${\ displaystyle N}$ ${\ displaystyle m}$ ${\ displaystyle M}$ ${\ displaystyle n}$ ${\ displaystyle p \ in N}$ ${\ displaystyle (U, \ phi)}$ ${\ displaystyle p}$

{\ displaystyle \ phi (N \ cap U) = (\ mathbb {R} ^ {n} \ times \ {0 \}) \ cap \ phi (U).}

Here, the as interpreted; the "0" on the right is the 0 of . Such cards are called cut cards. These define in a natural way a differentiable structure that is compatible with the differentiable structure of : If one identifies with , then the cut map is restricted to a map of and the set of all maps obtained in this way forms a differentiable atlas of . ${\ displaystyle \ mathbb {R} ^ {m}}$ ${\ displaystyle \ mathbb {R} ^ {n} \ times \ mathbb {R} ^ {mn}}$ ${\ displaystyle \ mathbb {R} ^ {mn}}$ ${\ displaystyle N}$ ${\ displaystyle M}$ ${\ displaystyle \ mathbb {R} ^ {n} \ times \ {0 \}}$ ${\ displaystyle \ mathbb {R} ^ {n}}$ ${\ displaystyle (U \ cap N, \ phi | _ {U \ cap N})}$ ${\ displaystyle (U, \ phi)}$ ${\ displaystyle U \ cap N}$ ${\ displaystyle N}$ ${\ displaystyle N}$

Whitney embedding theorem

Whitney's embedding theorem says that for every -dimensional differentiable manifold there is an embedding that identifies with a closed submanifold of the . The concept of the abstract differentiable manifold differs from that of the submanifold in, therefore, only in the intuition, but not in its mathematical properties. ${\ displaystyle n}$ ${\ displaystyle M}$ ${\ displaystyle M \ to \ mathbb {R} ^ {2n}}$ ${\ displaystyle M}$ ${\ displaystyle \ mathbb {R} ^ {2n}}$ ${\ displaystyle \ mathbb {R} ^ {2n}}$

classification

A topological manifold is a Hausdorff space that satisfies the second axiom of countability, together with an atlas. Under certain circumstances it is possible, for example by reducing the maps in the atlas, to obtain a differentiable atlas and thus to expand the topological manifold to a differentiable manifold. However, a differentiable structure cannot be found for every topological manifold. Under certain circumstances it is even possible to find non-equivalent differentiable atlases on a topological manifold. So there are also topological manifolds on which one can find different differentiable structures. From the perspective of differential geometry, there are then two different manifolds, while in topology it is only one object.

When classifying differentiable manifolds, one examines the question of how many different differentiable structures exist on a differentiable manifold. To put it more simply, one chooses a differentiable manifold, considers only the topological structure of this and examines how many different differentiable structures exist that make them a differentiable manifold. For differentiable manifolds of dimension smaller than four there is only one differentiable structure (apart from diffeomorphism). For all manifolds of dimension greater than four there are finitely many different differentiable structures. Manifolds of dimension four are exceptional in terms of differentiable structures. The simplest example of a non-compact four-dimensional differentiable manifold has uncountably many different differentiable structures, whereas the one with has exactly one differentiable structure. In the case of the four-dimensional sphere , in contrast to other "smaller" dimensions, it is not yet known how many differentiable structures it has. The following table contains the number of differentiable structures on the spheres up to dimension 12: ${\ displaystyle \ mathbb {R} ^ {4}}$ ${\ displaystyle \ mathbb {R} ^ {n}}$ ${\ displaystyle n \ neq 4}$

dimension	1	2	3	4th	5	6th	7th	8th	9	10	11	12
Number of differentiable structures	1	1	1	?	1	1	28	2	8th	6th	992	1

Infinite-dimensional manifolds

The manifolds presented here look locally like finite-dimensional space , so these manifolds are finite-dimensional by definition. ${\ displaystyle \ mathbb {R} ^ {n}}$

However, there are also several approaches in the literature to define infinitely dimensional differentiable manifolds. Usually one replaces the space in the definition with a locally convex topological vector space (the so-called model space), such as a Fréchet space , a Banach space or a Hilbert space . One then speaks of locally convex manifolds, Fréchet manifolds, Banach manifolds or Hilbert manifolds. Of course, such a definition only makes sense when one has agreed on how to define differentiable and mappings between infinite-dimensional locally convex spaces. While this is relatively uncritical for Banach spaces ( Fréchet derivation ), there are different, non-equivalent approaches for any locally convex spaces. ${\ displaystyle \ mathbb {R} ^ {n}}$ ${\ displaystyle C ^ {k}}$

Examples of infinite-dimensional manifolds:

the unit sphere in a Hilbert space is a -Hilbert manifold. ${\ displaystyle C ^ {\ infty}}$
the group of unitary operators on a Hilbert space is a -Banach manifold. ${\ displaystyle C ^ {\ infty}}$
the group of diffeomorphisms of the unit circle is a -Fréchet manifold. ${\ displaystyle C ^ {\ infty}}$

literature

John M. Lee: Introduction to Smooth Manifolds . 2nd Edition. Springer, New York 2003, ISBN 0-387-95448-1 (English).
R. Abraham, JE Marsden , T. Ratiu: Manifolds, Tensor Analysis, and Applications . 2nd Edition. Springer, Berlin 1988, ISBN 3-540-96790-7 (English).

Individual evidence

↑ ^a ^b M. I. Voitsekhovskii: Differentiable manifold . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, [ springer.de online]).
^ Felix Hausdorff: Collected works . Ed .: Egbert Brieskorn. Volume II: Fundamentals of set theory . Springer Verlag, Berlin et al. 2002, ISBN 3-540-42224-2 , pp. 72 .

[EoM-1] M. I. Voitsekhovskii: Differentiable manifold . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, [ springer.de online]).

[2] Felix Hausdorff: Collected works . Ed .: Egbert Brieskorn. Volume II: Fundamentals of set theory . Springer Verlag, Berlin et al. 2002, ISBN 3-540-42224-2 , pp. 72 .