---
tag: X7GP9BECD8
kind: concept
name: Tangential Cauchy–Riemann equations
msc: [32V05, 32V10, 35N10]
sources:
  - url: https://en.wikipedia.org/wiki/CR_manifold
---

## Summary

The tangential Cauchy–Riemann equations on a real submanifold of $\C^n$ are a restricted form of the [[Cauchy–Riemann equations|9PT772R94A]], see the [[definition|KQWJ8SGJEZ]].
They are taken only in the anti-holomorphic directions that are tangent to the submanifold.
Their solutions are the [[CR functions|JFKYYK2CEB]], and every restriction of a holomorphic function is one.
They form an overdetermined system of first-order linear partial differential equations.
Hans Lewy's study of them led to his example of a linear equation with no solutions.

## Explanations

### L1 human

A holomorphic function satisfies $\partial f / \partial \bar z_j = 0$ in every direction.
On a hypersurface, a function is only defined on the hypersurface, so only the derivatives along it make sense.
The tangential equations keep those combinations of the $\partial / \partial \bar z_j$ that point along the hypersurface.
On a real curve in $\C$ no such combination is left, so there are no equations at all; on a hypersurface in $\C^2$ there is one equation.