---
tag: 4WCJ49Z90S
kind: statement
role: theorem
name: Lichnerowicz formula
msc: [53C27, 58J60, 53C21]
uses: [QMRAVX14BJ, PYYQQKAZ0C, RSPZ9TCFQ8, 7CBT4SM83Y, ZHX2VYQDZ3, CAZYGE1T4E, BE9GTWMN0B, K0GVMM3YPF]
about: [QMRAVX14BJ, ZHX2VYQDZ3]
sources:
  - url: https://en.wikipedia.org/wiki/Lichnerowicz_formula
same_as:
  wikidata: Q6543306
  wikipedia: Lichnerowicz formula
---

## Statement

Let $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]] of dimension $n$ with a [[spin structure|RSPZ9TCFQ8]], and let $S$ be its [[spinor bundle|PYYQQKAZ0C]] with spin connection $\nabla$.
Let $D$ be the [[Dirac operator|QMRAVX14BJ]] of $M$, defined with Clifford multiplication satisfying $v \cdot v = -g(v, v)$.
Let $\nabla^* \nabla = -\sum_{i=1}^n \bigl(\nabla_{e_i} \nabla_{e_i} - \nabla_{\nabla_{e_i} e_i}\bigr)$ be the rough Laplacian, for a local orthonormal frame $e_1, \ldots, e_n$, and let $R$ be the [[scalar curvature|ZHX2VYQDZ3]] of $g$.
Then
$$D^2 \psi = \nabla^* \nabla \psi + \frac{R}{4} \, \psi$$
for every smooth section $\psi$ of $S$.

## Remarks

Work at a point with a frame satisfying $\nabla e_i = 0$ there.
Then $D^2 = \sum_{i,j} e_i e_j \nabla_{e_i} \nabla_{e_j}$, the terms with $i = j$ give $\nabla^* \nabla$, and the terms with $i \ne j$ give $\frac{1}{2} \sum_{i,j} e_i e_j R^S(e_i, e_j)$, where $R^S$ is the curvature of $S$.
The curvature of the spin connection is the Riemann tensor acting through Clifford multiplication, and the first Bianchi identity reduces this sum to $\frac{R}{4}$.

Example: on flat $\R^n$, $R = 0$ and the formula says that $D^2 = -\sum_j \partial_j^2$ on spinor fields, the defining property of the flat Dirac operator.
On the round sphere $S^n$ of radius $1$, $R = n(n-1) > 0$, so $D^2 \ge \frac{n(n-1)}{4}$ as an operator, which bounds the eigenvalues of $D$ away from $0$.

## Explanations

### L1 human

The square of the Dirac operator is a Laplacian plus one extra term, and that term is a quarter of the scalar curvature.
So the Dirac operator feels the curvature of the manifold only through one number at each point.
When that number is positive, it pushes the square of the operator up and leaves no room for spinors killed by it.

### L2 human

The formula is a special case of the Bochner–Weitzenböck formula for the Dirac operator of a [[Dirac bundle|K0GVMM3YPF]], like the [[Weitzenböck identity|BE9GTWMN0B]] for forms.
Integrating it on a closed manifold gives $\norm{D\psi}^2 = \norm{\nabla\psi}^2 + \frac{1}{4} \int_M R \abs{\psi}^2$, which is the source of the Lichnerowicz vanishing theorem.
For a spinor bundle twisted by a bundle $E$ with connection, an extra term appears: Clifford multiplication by the curvature of $E$.