nLab
inner product on vector bundles

Context

Bundles

Linear algebra

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

Contents

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

  2. XX be a topological space,

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

Then an inner product on EE is

such that

  • for each point xXx \in X the function

    ,| x:E xE x \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 XX be a paracompact Hausdorff space. Then on every topological vector bundle EXE \to X on XX there exists an inner product of topological vector bundles (def. 1)

(e.g. Hatcher, prop. 1.2)

Proof

Let {U iX} iI\{U_i \subset X\}_{i \in I} be an open cover of XX over which EXE \to X admits a local trivialization

{ϕ i:U i× nE| U i} iI. \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:X[0,1]} iI. \{f_i \;\colon\; X \to [0,1]\}_{i \in I} \,.

Write

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

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

By the compact support of f if_i inside U iXU_i \subset X, the functions

, i : E| U i XE| U i ϕ i 1 U iϕ i 1 (U i× n) U i(U i× n) U i×( n n) U i× ((x,i),(v 1,v 2)) AAA ((x,i),f i(x)v 1,v 2 n). \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 XEE \otimes_X E, which we denote by the same symbol , i:E XEX×\langle -,-\rangle_i \colon E \otimes_X E \to X \times \mathbb{R}.

We then claim that the sum

,:iI, i:E XEX× \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 if_i. Hence it is pointwise a finite sum of positive definite symmetic bilinear forms on E xE_x, and as such itself pointwise a positive definite symmetric bilinear form.

References

  • Glenys Luke, Alexander S. Mishchenko, Vector bundles and their applications, Math. and its Appl. 447, Kluwer 1998. viii+254 pp. 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, (partly finished book) web

Revised on September 5, 2017 14:34:49 by Urs Schreiber (77.56.177.247)