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:
which assigns to each point p a positive definite symmetric bilinearform.
And the tensor bundles are of the form:
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 ??