The Bochner Vanishing Theorem

The Bochner technique uses curvature to constrain harmonic forms, establishing vanishing theorems for cohomology. It is a powerful method linking curvature to topology.

The Bochner Formula

Theorem10.6Weitzenbock formula

On a Riemannian manifold $(M, g)$ , the Hodge Laplacian and the connection Laplacian (rough Laplacian) on $k$ -forms are related by

\Delta\omega = \nabla^*\nabla\omega + \mathcal{R}_k(\omega),

where $\nabla^*\nabla = -\mathrm{tr}(\nabla^2)$ is the connection Laplacian and $\mathcal{R}_k$ is a zeroth-order curvature term involving the Riemann tensor. For 1-forms: $\mathcal{R}_1(\omega) = \mathrm{Ric}(\omega^\sharp, \cdot)^\flat$ , giving the Bochner formula:

\Delta\omega = \nabla^*\nabla\omega + \mathrm{Ric}(\omega).

The Vanishing Theorem

Theorem10.7Bochner vanishing theorem

Let $(M^n, g)$ be a compact oriented Riemannian manifold.

If $\mathrm{Ric} > 0$ everywhere, then $\mathcal{H}^1(M) = 0$ , so $H^1_{\mathrm{dR}}(M) = 0$ (equivalently, $b_1 = 0$ ).
If $\mathrm{Ric} \geq 0$ everywhere, then every harmonic 1-form is parallel: $\nabla\omega = 0$ .

Proof

Let $\omega$ be a harmonic 1-form: $\Delta\omega = 0$ . By the Bochner formula:

0 = \Delta\omega = \nabla^*\nabla\omega + \mathrm{Ric}(\omega).

Taking the $L^2$ inner product with $\omega$ :

0 = \langle \nabla^*\nabla\omega, \omega \rangle + \langle \mathrm{Ric}(\omega), \omega \rangle = |\nabla\omega|^2_{L^2} + \int_M \mathrm{Ric}(\omega^\sharp, \omega^\sharp) \, \mathrm{dvol}_g.

Both terms are non-negative when $\mathrm{Ric} \geq 0$ . If $\mathrm{Ric} > 0$ , the second term is strictly positive unless $\omega = 0$ , so $\omega = 0$ and $\mathcal{H}^1 = 0$ . If $\mathrm{Ric} \geq 0$ , both terms must vanish: $\nabla\omega = 0$ (parallel) and $\mathrm{Ric}(\omega^\sharp, \omega^\sharp) = 0$ . $\blacksquare$

■

Consequences and Generalizations

ExampleTopological consequences

If $(M^n, g)$ is compact with $\mathrm{Ric} > 0$ , then $b_1(M) = 0$ . Combined with Bonnet-Myers ( $\pi_1$ finite) and the universal coefficient theorem: $H_1(M; \mathbb{Z})$ is finite.
If $(M^n, g)$ is compact with $\mathrm{Ric} \geq 0$ , then $b_1(M) \leq n$ , with equality if and only if $M$ is a flat torus.
The flat torus $T^n$ achieves $b_1 = n$ with $\mathrm{Ric} = 0$ : the $n$ coordinate 1-forms $dx^1, \ldots, dx^n$ are parallel (hence harmonic).

Theorem10.8Bochner technique for higher forms

The Weitzenbock formula for $k$ -forms involves the full curvature operator $\mathcal{R}_k$ . Specific results:

If the curvature operator $\mathcal{R}: \Lambda^2 \to \Lambda^2$ is positive definite, then $b_k(M) = 0$ for $1 \leq k \leq n-1$ (Gallot-Meyer theorem).
On a Kahler manifold with positive bisectional curvature, $H^{p,0}(M) = 0$ for $p \geq 1$ (the Kodaira vanishing theorem follows from this approach).

RemarkThe Bochner method in broader context

The Bochner technique -- integrating a Weitzenbock identity to extract global information from local curvature -- is one of the most versatile methods in geometric analysis. It extends to: harmonic maps (Eells-Sampson), spinors (Lichnerowicz's proof that manifolds with $S > 0$ have no harmonic spinors, giving obstructions to positive scalar curvature), and holomorphic sections of line bundles (Kodaira vanishing).