(see also Chern-Weil theory, parameterized homotopy theory)
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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
$k$ be a topological field (such as the real numbers or complex numbers with their Euclidean metric topology ),
$X$ be a topological space,
$E \to X$ a topological vector bundle over $X$ (over $\mathbb{R}$, say).
Then an inner product on $E$ is
a vector bundle homomorphism
from the tensor product of vector bundles of $E$ with itself to the trivial line bundle
such that
for each point $x \in X$ the function
is an inner product on the fiber vector space, hence a positive-definite symmetric bilinear form.
Let $X$ be a paracompact Hausdorff space. Then on every topological vector bundle $E \to X$ on $X$ there exists an inner product of topological vector bundles (def. 1)
(e.g. Hatcher, prop. 1.2)
Let $\{U_i \subset X\}_{i \in I}$ be an open cover of $X$ over which $E \to X$ 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 $\mathbb{R}^n$.
By the compact support of $f_i$ inside $U_i \subset X$, the functions
extend by zero to continuous functions on all of $E \otimes_X E$, which we denote by the same symbol $\langle -,-\rangle_i \colon E \otimes_X E \to X \times \mathbb{R}$.
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 $f_i$. Hence it is pointwise a finite sum of positive definite symmetic bilinear forms on $E_x$, 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, (partly finished book) web
Last revised on September 5, 2017 at 14:34:49. See the history of this page for a list of all contributions to it.