# Vector bundles with inner products induced from the metric

If $$E \to M$$ is a tensor bundle over a Riemannian manifold (M, g) then the metric $$g$$ induces an inner product on the fibers of $$E$$ in the same way that one can give tensor products of inner product spaces an inner product.

We can also consider direct sums of tensor bundles which will also have an inner product induced from the metric in the same way that the direct sum of some inner product spaces is an inner product space with the inner product taken to be the sum of the inner products on the projections.

In this way one can build a collection of vector bundles over $$M$$ which have some sort of natural inner product structure induced from the metric $$g$$. Does this collection of vector bundles have a name and is this talked about in any texts?

• I think you mean the various associated bundles (coming from various representations of the orthogonal group defined by $g$). – Ted Shifrin Mar 23 at 1:19