Bochner's theorem (Riemannian geometry)

In mathematics, Salomon Bochner proved in 1946 that any Killing vector field of a compact Riemannian manifold with negative Ricci curvature must be zero.

[1][2][3] The theorem is a corollary of Bochner's more fundamental result which says that on any connected Riemannian manifold of negative Ricci curvature, the length of a nonzero Killing vector field cannot have a local maximum.

In particular, on a closed Riemannian manifold of negative Ricci curvature, every Killing vector field is identically zero.

However, on a Riemannian metric of negative Ricci curvature, the right-hand side is strictly positive wherever X is nonzero.

Since Killing vector fields on connected manifolds are uniquely determined from their value and derivative at a single point, it follows that X must be identically zero.