---
tag: CK1DCPC3D3
kind: concept
name: Hermitian manifold
msc: [53C55, 53B35]
specializes: [7CBT4SM83Y]
sources:
  - url: https://en.wikipedia.org/wiki/Hermitian_manifold
same_as:
  wikidata: Q5741935
  wikipedia: Hermitian manifold
---

## Summary

A Hermitian manifold is a complex manifold with a Riemannian metric that is compatible with multiplication by $i$ on tangent vectors.
It is the complex analogue of a Riemannian manifold, and every complex manifold admits such a metric.
Each Hermitian metric has a fundamental $2$-form, and when this form is closed the manifold is a Kähler manifold.
Kähler manifolds are the setting of Hodge theory and of much of complex algebraic geometry.

## Explanations

### L1 human

On a complex manifold one can turn each tangent vector by a quarter turn, using multiplication by $i$.
A Hermitian metric measures lengths and angles so that this quarter turn never changes a length or an angle.
The plane $\C$ with the usual distance is the simplest example: rotating an arrow by $90$ degrees keeps its length.
Pairing the metric with the quarter turn gives a $2$-form, which measures oriented areas.