nLab inner product on vector bundles

Idea

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.

Definition

Definition

(inner product on topological vector bundles)

Let

1. $k$ be a topological field (such as the real numbers or complex numbers with their Euclidean metric topology ),

2. $X$ be a topological space,

3. $E \to X$ a topological vector bundle over $X$ (over $\mathbb{R}$, say).

Then an inner product on $E$ is

• a vector bundle homomorphism

  $$\langle -,-\rangle \;\colon\; E \otimes_X E \longrightarrow X \times \mathbb{R}$$

  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

  $$\langle -,-\rangle|_x \colon E_x \otimes E_x \to \mathbb{R}$$

  is an inner product on the fiber vector space, hence a positive-definite symmetric bilinear form.

Properties

Existence

Proposition

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)

Proof

Let $\{U_i \subset X\}_{i \in I}$ be an open cover of $X$ over which $E \to X$ admits a local trivialization

$$\left\{ \phi_i \;\colon\; U_i \times \mathbb{R}^n \overset{\simeq}{\longrightarrow} E|_{U_i} \right\}_{i \in I} \,.$$

Since paracompact Hausdorff spaces equivalently admit subordinate partitions of unity we may choose a partition of unity

$$\{f_i \;\colon\; X \to [0,1]\}_{i \in I} \,.$$

Write

$$\langle -,-\rangle_{\mathbb{R}^n} \;\colon\; \mathbb{R}^n \otimes \mathbb{R}^n \longrightarrow \mathbb{R}$$

for the standard inner product on $\mathbb{R}^n$.

By the compact support of $f_i$ inside $U_i \subset X$, the functions

$$\array{ \langle -,-\rangle_i &\colon& E|_{U_i} \otimes_X E|_{U_i} &\overset{ \phi_i^{-1} \otimes_{U_i} \phi_i^{-1} }{\longrightarrow}& (U_i \times \mathbb{R}^n ) \otimes_{U_i} (U_i \times \mathbb{R}^n) &\overset{\simeq}{\to}& U_i \times (\mathbb{R}^n \otimes \mathbb{R}^n) &\overset{}{\longrightarrow}& U_i \times \mathbb{R} \\ && && && ((x,i),(v_1,v_2)) & \overset{\phantom{AAA}}{\mapsto} & ((x,i), f_i(x) \cdot \langle v_1, v_2\rangle_{\mathbb{R}^n} ) } \,.$$

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

$$\langle -,-\rangle \;\colon\; \underset{i \in I}{\sum} \langle -,-\rangle_i \;\colon\; E \otimes_X E \longrightarrow X \times \mathbb{R}$$

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.

Related concepts

• topological Hermitian K-theory

• vector bundle

  • real vector bundle

  • complex vector bundle

    • holomorphic vector bundle, pseudoholomorphic vector bundle

  • universal vector bundle

  • dual vector bundle

  • module bundle

  • direct sum of vector bundles

  • short exact sequence of vector bundles

  • connection on a vector bundle

  • flat vector bundle

  • real vector bundle, complex vector bundle

  • super vector bundle

References

• Glenys Luke, Alexandr S. Mishchenko, Vector bundles and their applications, Math. and its Appl. 447 Kluwer (1998) [doi:10.1007/978-1-4757-6923-4, MR99m:55019]

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)