nLab inner product on vector bundles




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Basic facts




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)


  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.




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. )

(e.g. Hatcher, prop. 1.2)


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} \,.


, 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.


  • 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 (webpage, pdf)

Last revised on April 1, 2021 at 11:20:29. See the history of this page for a list of all contributions to it.