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The concept of inner product on vector bundles is the evident generalization of that of inner product on vector spaces to vector bundles: a fiber-wise inner product of vector spaces.
(inner product on topological vector bundles)
Let
be a topological field (such as the real numbers or complex numbers with their Euclidean metric topology ),
be a topological space,
a topological vector bundle over (over , say).
Then an inner product on is
a vector bundle homomorphism
from the tensor product of vector bundles of with itself to the trivial line bundle
such that
for each point the function
is an inner product on the fiber vector space, hence a positive-definite symmetric bilinear form.
Let be a paracompact Hausdorff space. Then on every topological vector bundle on there exists an inner product of topological vector bundles (def. )
(e.g. Hatcher, prop. 1.2)
Let be an open cover of over which admits a local trivialization
Since paracompact Hausdorff spaces equivalently admit subordinate partitions of unity we may choose a partition of unity
Write
for the standard inner product on .
By the compact support of inside , the functions
extend by zero to continuous functions on all of , which we denote by the same symbol .
We then claim that the sum
is an inner product as required. Notice that the sum is well defined by local finiteness of the supports of the partition functions . Hence it is pointwise a finite sum of positive definite symmetic bilinear forms on , and as such itself pointwise a positive definite symmetric bilinear form.
Discussion with an eye towards topological K-theory is in
Max Karoubi, K-theory. An introduction, Grundlehren der Mathematischen Wissenschaften 226, Springer 1978. xviii+308 pp.
Allen Hatcher, section 1.1 of: Vector bundles and K-Theory (webpage, pdf)
Last revised on October 26, 2023 at 12:14:06. See the history of this page for a list of all contributions to it.