
Tensor bundle
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

Consider a vector space V with an inner product $\displaystyle \langle , \rangle$ defined. The inner product induces a canonical isomorphism $\displaystyle \phi$ between V and V*: $\displaystyle (\phi(v))(w) = \langle v, w \rangle $. So we can identify V and V* by this isomorphism. Then we can consider only tensor products $\displaystyle V_{r,0}$.
Define an inner product on $\displaystyle V_{r,0}$ as follows:
$\displaystyle \langle v_1 \otimes v_2 \otimes ... \otimes v_r, w_1 \otimes w_2 \otimes ... \otimes w_r \rangle = \langle v_1, w_1 \rangle \langle v_2, w_2 \rangle ... \langle v_r, w_r \rangle$

Excuse me this was a bad mistake of me! In europe it is very early in the morning! ;)
Regards

why did you say that I was considering only for even ranks?

Excuse me! You are absolutely right. I did a mistake. I have forgotten that the inner product takes two elements of the tensor bundle therefor you are right!
Now i have to show that this construction is indeed continuous. Why do we have this property?
Regards

Since the induced inner product is (smoothly) expressible by the original one g, and g is smooth, we're done.
You can write down the exact expression to be sure.

Thank you again!
I think they are continuous, since our g is the product of the original one g (which is smooth).
So now we have defined a smooth inner product on each Tensor space of the form (r,s).(with r,s fixed).
What does this construction mean? What can i do with it? Or better: What do we need it for?
Do you know perhaps what this meaning of this construction is?
Regards