Differentiable Manifold - Definition

Definition

A presentation of a topological manifold is a second countable Hausdorff space that is locally homeomorphic to a linear space, by a collection (called an atlas) of homeomorphisms called charts. The composition of one chart with the inverse of another chart is a function called a transition map, and defines a homeomorphism of an open subset of the linear space onto another open subset of the linear space. This formalizes the notion of "patching together pieces of a space to make a manifold" – the manifold produced also contains the data of how it has been patched together. However, different atlases (patchings) may produce "the same" manifold, and, on the converse, a manifold does not come with a preferred atlas. And, thus, one defines a topological manifold to be a space as above with an equivalence class of atlases, where one defines equivalence of atlases below.

There are a number of different types of differentiable manifolds, depending on the precise differentiability requirements on the transition functions. Some common examples include the following.

• A differentiable manifold is a topological manifold equipped with an equivalence class of atlases whose transition maps are all differentiable. In broader terms, a Ck-manifold is a topological manifold with an atlas whose transition maps are all k-times continuously differentiable.

• A smooth manifold or C∞-manifold is a differentiable manifold for which all the transition maps are smooth. That is, derivatives of all orders exist; so it is a Ck-manifold for all k. An equivalence class of such atlases is said to be a smooth structure.

• An analytic manifold, or Cω-manifold is a smooth manifold with the additional condition that each transition map is analytic: the Taylor expansion is absolutely convergent and equals the function on some open ball.

• A complex manifold is a topological space modeled on a Euclidean space over the complex field and for which all the transition maps are holomorphic.

While there is a meaningful notion of a Ck atlas, there is no distinct notion of a Ck manifold other than C0 (continuous maps: a topological manifold) and C∞ (smooth maps: a smooth manifold), because every Ck-structure with k > 0, there is a unique Ck-equivalent C∞-structure (every Ck-structure is uniquely smoothable) – a result of Whitney (and further, two Ck atlases that are equivalent to a single C∞ atlas are equivalent as Ck atlases, so two distinct Ck atlases do not collide); see Differential structure: Existence and uniqueness theorems for details. Thus one uses the terms "differentiable manifold" and "smooth manifold" interchangeably. This is in stark contrast to Ck maps, where there are meaningful differences for different k. For example, the Nash embedding theorem states that any manifold can be Ck isometrically embedded in Euclidean space RN – for any 1 ≤ k ≤ ∞ there is a sufficiently large N, but N depends on k.

On the other hand, complex manifolds are significantly more restrictive. As an example, Chow's theorem states that any projective complex manifold is in fact a projective variety – it has an algebraic structure.