Differential structure

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In mathematics, an n-dimensional differential structure (or differentiable structure) on a set M makes it into an n-dimensional differential manifold, which is a topological manifold with some additional structure that allows us to do differential calculus on the manifold. If M is already a topological manifold, we require that the new topology is identical to the existing one.

Definition

For a natural number n and some k which may be a non-negative integer or infinity, an n-dimensional C^k differential structure is defined using a C^k-atlas, which is a set of bijections called charts between a set of subsets of M (whose union is the whole of M), and a set of open subsets of an n-dimensional vector space:

<math>\varphi_{i}:W_{i}\subset M\rightarrow U_{i}\subset\mathbb{R}^{n}.</math>

which are C^k-compatible (in the sense defined below):

Each such map provides a way in which certain subsets of the manifold may be viewed as being like open subsets of <math>\mathbb{R}^{n}</math> but the usefulness of this notion depends on to what extent these notions agree when the domains of two such maps overlap.

Consider two charts:

<math>\varphi_{i}:W_{i}\rightarrow U_{i}</math>,

<math>\varphi_{j}:W_{j}\rightarrow U_{j}</math>.

The intersection of the ranges of these two functions is:

<math>W_{ij}=W_{i}\cap W_{j}\;</math>

and is mapped to two images

<math>U_{ij}=\varphi_{i}\left(W_{ij}\right)</math>,

<math>U_{ji}=\varphi_{j}\left(W_{ij}\right)</math>

by the two chart maps.

The transition map between the two charts is the map between the two images of this intersection under the two chart maps.

<math>\varphi_{ij}:U_{ij}\rightarrow U_{ji}</math>

<math>\varphi_{ij}(x)=\varphi_{j}\left(\varphi_{i}^{-1}\left(x\right)\right).</math>

Two charts <math>\varphi_{i},\,\varphi_{j}</math> are C^k-compatible if

<math>U_{ij},\, U_{ji}</math>

are open, and the transition maps

<math>\varphi_{ij},\,\varphi_{ji}</math>

have continuous derivatives of order k. If k = 0, we only require that the transition maps are continuous, consequently a C⁰-atlas is simply another way to define a topological manifold. If k = ∞, derivatives of all orders must be continuous. A family of C^k-compatible charts covering the whole manifold is a C^k-atlas defining a C^k differential manifold. Two atlases are C^k-equivalent if the union of their sets of charts forms a C^k-atlas. In particular, a C^k-atlas that is C⁰-compatible with a C⁰-atlas that defines a topological manifold is said to determine a C^k differential structure on the topological manifold. The C^k equivalence classes of such atlases are the distinct C^k differential structures of the manifold.

Existence and uniqueness theorems

On any manifold with a C^k structure for k>0, there is a unique C^k-compatible C^∞-structure, a theorem due to Whitney. On the other hand, there exist topological manifolds which admit no differential structures, see Donaldson's theorem (confer Hilbert's fifth problem).

When people count differential structures on a manifold, they usually count them modulo orientation-preserving homeomorphisms. There is only one differential structure of any manifold of dimension smaller than 4. For all manifolds of dimension greater than 4 there is a finite number of differential structures on any compact manifold. There is only one differential structure on <math> \mathbb{R}^{n}</math> except when <math>n = 4</math>, in which case there are uncountably many.

Differential structures on spheres of dimensions from 1 to 18

The following table lists the numbers of differential structures (modulo orientation-preserving homeomorphism) on the <math>n</math>-sphere for dimensions <math>n</math> up to dimension 18. Spheres with differential structures different from the usual one are known as exotic spheres.

It is not currently known how many differential structures there are on the 4-sphere, beyond that there is at least one. There may be one, a finite number, or an infinite number. The claim that there is just one is known as the smooth Poincaré conjecture.

See also

• Atlas
• Manifoldde:Differentialstruktur