# Differentiable Manifolds/Vector (sub-)bundles, sections, foliations, distributions and Frobenius' theorem

## What are vector bundles?[edit | edit source]

**Definition 10.1**:

Let be a manifold of class , where as usual. A **vector bundle of ** is a manifold of class together with

- a function (which we shall call
**projection of**) such that for each , is a finite-dimensional vector space over the real numbers AND - for each an open set with and such that there is a diffeomorphism of the product manifold to the manifold of class (which shall be called
**bundle chart of**), where is the dimension of , such that for each the function has as its image and is a linear isomorphism.

**Lemma 10.2**: Let be a vector bundle of the manifold with projection . Then if are contained in the same connected component of , the dimensions of and are equal.

**Proof**:

We define the function . Let now be contained in the connected component of . If we pick any point , due to the definition of a vector bundle, there is an open set with and a bundle chart , where is the dimension of , such that has as its image and is a linear isomorphism. Therefore, for all , the dimension of is equal to .

From this follows that the set and its complement in are both open (since for every point in the complement of this set there also exists an open neighbourhood such that all points in this neighbourhood satisfy and thus all the points in that neighbourhood are also contained in the complement).

But if the set and its complement in are both open, then so are the respective intersections with with respect to the subspace topology on induced by the topology of . But, by definition of a connected component, is a connected set with respect to the subspace topology, and thus, since is open and closed (since the complement is open) and nonempty (as it contains ), this set equals the whole set by definition of connected sets.

## The tangent and cotangent bundles as part of a respective vector bundle[edit | edit source]

In the following section we want to show, that both tangent and cotangent bundle with a specific atlas are manifolds, and if we define specific projections and bundle charts, we obtain vector bundles out of them and the tangent and cotangent bundles.

**Definition 10.3**:

Let be a manifold and let be an open subset of . Then we define:

**Theorem 10.4**:

Let be a -dimensional manifold of class , where , with atlas . For each , we define the function

and the set

Further, we define the topology on to be

This is a topology and is an atlas of class and together with is a -dimensional manifold of class .

**Theorem 10.5**:

Let be equipped with the atlas as defined in the statement of theorem 10.4. If we define

and for each

- ,

then with as projection and the functions is a vector bundle.

**Definition 10.6**:

Let be a manifold and let be an open subset of . Then we define:

**Theorem 10.7**:

Let be a -dimensional manifold of class .