the Bochner-Weitzenböck formula
/ BOKH-ner VITE-sen-burk /
There are two natural ways to build a Laplacian on differential forms: the analytic one from differentiation and integration (the Hodge Laplacian), and the geometric one from the connection (the rough Laplacian). On functions they agree, but on forms they disagree — and the precise amount by which they differ is curvature. The Bochner-Weitzenbock formula is the exact bookkeeping of that difference. It is the single identity that lets curvature bounds force topological conclusions, the linchpin of the 'Bochner technique'.
On a Riemannian manifold the rough (or connection) Laplacian on a tensor or form is nabla^* nabla = -trace(nabla^2), the negative trace of the second covariant derivative. The Hodge Laplacian is Delta = d d^* + d^* d. The Bochner-Weitzenbock formula says Delta = nabla^* nabla + R, where R is a zeroth-order curvature term — a bundle endomorphism built algebraically from the curvature tensor. For 1-forms the formula is the Bochner identity Delta omega = nabla^* nabla omega + Ric(omega), where Ric is the Ricci curvature acting on the form. The technique that follows is beautiful and short: suppose omega is harmonic, Delta omega = 0. Pair the formula with omega and integrate over a compact manifold; the rough-Laplacian term contributes integral |nabla omega|^2 (nonnegative), so 0 = integral |nabla omega|^2 + integral Ric(omega, omega). If Ricci is nonnegative, both terms are nonnegative and must each vanish, forcing nabla omega = 0 (omega is parallel) and, if Ricci is somewhere strictly positive, omega = 0. Hence positive Ricci kills harmonic 1-forms, so b_1 = 0.
This is the prototype of how analysis converts a curvature hypothesis into a topological theorem, and the same scheme (Weitzenbock for the Dirac operator, the spinor Laplacian, etc.) underlies the Lichnerowicz vanishing theorem and much of index theory. The honest caveats. First, the curvature term and its sign depend on degree and convention: for 1-forms it is Ricci, but for higher-degree forms it is a more complicated curvature operator (involving the full curvature tensor, not just Ricci), so 'positive Ricci kills harmonic forms' is true in degree one but not automatically in higher degrees. Second, the conclusion needs compactness for the integration-by-parts step; on noncompact manifolds the boundary terms must be controlled. Third, this is genuinely a Ricci (not sectional) statement in degree one, illustrating the general principle that Ricci bounds are strictly weaker than sectional bounds.
On a compact manifold with strictly positive Ricci curvature, the Bochner argument shows there are no nonzero harmonic 1-forms, hence the first Betti number b_1 vanishes — for instance the round sphere S^n with n >= 2 has b_1 = 0, recovered analytically rather than topologically.
Positive Ricci curvature forces b_1 = 0 via the Bochner technique — a curvature-to-topology bridge.
The curvature term in degree one is Ricci, but in higher degrees it is the full curvature operator, so do not blindly extend 'positive Ricci kills harmonic forms' beyond 1-forms; the sign also flips with the curvature convention.