**Defition of Topological manifold**

Suppose is a topological space, for every point in , there exists a neighborhood of and such that is homoemorphism of and an open subset in , then is called dimensional topological manifold.

In order to have differentiation and integration on manifold, we need have differentiable structure.

Suppose is dimensional manifold. If a given set of coordinate chars satisfies

(1) coves

(2) the transition map between two local coordinate charts and are smooth. Namely the map

is differentiable.

(3) is maximal. If there exists some such that it is compatible to every , then .

**Tangent vector on .**

A tangent vector at on is a map satisfies

(1) , for and are real numbers;

(2) .

Suppose , then induces a tangent vector at by

If we have a coordinate near , then

where . Then

is a tangent vector at .

**Tangent space **

All tangent vectors form a linear space. For , there exists such that

where . Then any tangent vector , we have

Moreover we can prove that is linearly independent. So forms a basis of the tangent space.

**Tangent bundle **

We can assign topology and differentiable stucture on tangent bundle.

Suppose is dimensional manifold. Then

For every , there exists a coordinates of on , say , then Let , define

where . Obviously, is bijective. Define the open set of is . So is a dimensional manifold, and is continuous.

Next suppose has differentiable structure. There exists an open cover , then has an open cover and for

where

Hence is transition function. has a differentiable structure.

**Metric **

Similarly is the cotangent bundle which is also a manifold. Riemanian metric is a smooth function

satisfies

(1)

(2) matrix