Hello,

i try to show that one can define a metric on all tensor bundles with the riemannian metric.

I have really no idea, how i can show this. Here are the definition, which i know:

A Riemannian Metric on a Manifold M is a map f which assigns to each point p this:

$\displaystyle f(p):T_{p}M\;\times\;T_{p}M->\mathbb{R}$ which assigns to each point p a positive definite symmetric bilinearform.

And the tensor bundles are of the form:

$\displaystyle T_{r,s}(M):=\bigsqcup_{m\in M}(T_{m}M)_{r,s}$

with $\displaystyle (T_{m}M)_{r,s}:=T_{m}M\otimes...\otimes T_{m}M\otimes (T_{m}M)^{*}\otimes...\otimes (T_{m}M)^{*}$

Ok, meanwhile i'm not sure what the word "metric" mean. A metric on all tensor bundles. Is it a metric in the usual sense, that is a metric space?

Or is it a metric like the riemannian metric is. I.e. a positive definite symmetric bilinearform ??

Regards