(see also Chern-Weil theory, parameterized homotopy theory)
vector bundle, (∞,1)-vector bundle
topological vector bundle, differentiable vector bundle, algebraic vector bundle
direct sum of vector bundles, tensor product of vector bundles, inner product of vector bundles?, dual vector bundle
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
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. )
(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.