Bundle with Metric

Metric on a Real Vector bundle

A metric on a vector bundle $\xi$ is defined on the pullback bundle.

$$g:E{\times }_{B}E\to \mathbb{R}$$

such that $\forall b\in B,{\pi }^{-1}b\times {\pi }^{-1}b\to \mathbb{R}$ is an inner product.

Using this bundle metric, we can define the norm on the bundle

$$\Vert \cdot \Vert :E\to {\mathbb{R}}_{\ge 0}.$$

such that $v\mapsto \sqrt{g(v,v)}$.

A bundle with metric is often callled a Euclidean vector bundle.

Examples

We have Riemannian Manifold as a example.

1. If $M^n\subset {\mathbb{R}}^N$ we can use the inner product inherited from ${\mathbb{R}}^N$, so it is a Riemannian manifold.

2. Trivial bundle has a metric

   which looks like $((b,v),(b,w))\mapsto v\cdot w$.

3. If $M^n\subset {\mathbb{R}}^N$, $TM=\left\{(x,v)\in M\times {\mathbb{R}}^N\mid x=\gamma (0),v={\gamma }^{\prime }(0)\right\}$ we have $\Vert (x,v)\Vert =\Vert v\Vert ,g((x,v),(x,w))=v\cdot w$.

Exercise.