---
tag: Q6X6W2N29G
kind: concept
name: Hermitian metric on a vector bundle
msc: [32L05, 53C55]
sources:
  - url: https://en.wikipedia.org/wiki/Hermitian_manifold
---

## Summary

A Hermitian metric on a complex [[vector bundle|P4DZM975BV]] is a choice of [[Hermitian inner product|0NB0AWGJBW]] on each fibre that varies smoothly with the base point.
It lets one measure the length of sections and define the Chern connection and its curvature.
Every complex vector bundle on a paracompact manifold has one, by a partition of unity.
A Hermitian metric on the holomorphic tangent bundle of a complex manifold is what makes it a [[Hermitian manifold|CK1DCPC3D3]].

## Explanations

### L1 human

Each fibre of a complex vector bundle is a complex vector space, but it comes with no notion of length.
A Hermitian metric gives each fibre an inner product, as on $\C^r$, and asks that it change smoothly from fibre to fibre.
Then each vector in the bundle has a length, and each section has a length function on the base.
On the trivial line bundle over $\C$, the rule $\abs{v}_z^2 = e^{-\abs{z}^2} \abs{v}^2$ is such a metric: it is the usual length, rescaled by a positive smooth factor at each point.