the Bochner technique
/ BOKH-ner /
Sometimes you can prove a geometric object simply does not exist by writing down an integral that, under a curvature assumption, can only balance if the object was zero all along. The Bochner technique is exactly this kind of argument: a way of forcing a tensor or differential form to vanish by combining a curvature lower bound with an integration-by-parts identity. It is the standard machine for converting 'positive curvature' into 'no harmonic forms,' and hence into constraints on a manifold's topology.
Here is the prototype, Bochner's original theorem. On a compact oriented Riemannian manifold, the Hodge Laplacian on 1-forms factors through a Weitzenboeck (Bochner) identity: Delta = nabla^* nabla + Ric, where nabla^* nabla is the (nonnegative) rough Laplacian and Ric is the Ricci curvature acting on the form. Now take a harmonic 1-form omega (Delta omega = 0). Integrate the identity paired with omega over M: the left side is zero, the rough-Laplacian term integrates to the nonnegative integral of |nabla omega|^2, and the curvature term integrates to the integral of Ric(omega, omega). If Ric >= 0 everywhere, both nonnegative terms must vanish, forcing nabla omega = 0 (omega is parallel) and Ric(omega, omega) = 0; if moreover Ric > 0 somewhere, omega must be identically zero. Since harmonic 1-forms represent first de Rham cohomology (Hodge theory), this kills H^1: a compact manifold with Ric >= 0 has first Betti number at most n, and with Ric > 0 it has b_1 = 0.
The Bochner technique is one of the most flexible tools in geometric analysis: the same recipe (Weitzenboeck identity + integration + curvature sign) proves vanishing of harmonic forms of other degrees, Killing fields, holomorphic forms on Kahler manifolds, and even underlies Lichnerowicz's obstruction to positive-scalar-curvature spin metrics via the Dirac operator. Honest caveats. First, it needs the integration-by-parts to have no boundary term — hence compactness (or suitable decay); on noncompact manifolds extra hypotheses are required and the splitting theorem's use of Bochner is much more delicate. Second, the conclusion is a vanishing/parallel statement, not a classification, and it sees only the real cohomology that harmonic forms detect — torsion is invisible to it. Third, the precise curvature term depends on the degree and the Weitzenboeck formula in question; do not assume 'Ricci' is always the right curvature.
Take the flat torus T^n = R^n / Z^n: it has Ric = 0, so Bochner says every harmonic 1-form is parallel — and indeed the parallel forms dx^1, ..., dx^n span an n-dimensional space, giving first Betti number b_1 = n, the maximum the Ric >= 0 case allows. Now perturb to ANY metric with Ric > 0 somewhere and Ric >= 0 everywhere: Bochner forces b_1 = 0, so the manifold cannot have a torus factor or any nontrivial H^1. This is why, for instance, the round sphere S^n (Ric > 0) has b_1 = 0 — there are no nonzero harmonic 1-forms to support a 1-cycle.
Ric >= 0 makes harmonic 1-forms parallel (torus: b_1 = n); Ric > 0 kills them outright (sphere: b_1 = 0).
Bochner sees only the real cohomology that harmonic forms represent; it is blind to torsion in integral cohomology. And it needs no boundary term in the integration — compactness (or decay) is essential, not cosmetic.