Splitting theorem

In the mathematical field of differential geometry, there are various splitting theorems on when a pseudo-Riemannian manifold can be given as a metric product.

[1] In 1971, Jeff Cheeger and Detlef Gromoll proved that if a geodesically complete and connected Riemannian manifold of nonnegative Ricci curvature contains any geodesic line, then it must split isometrically as the product of a complete Riemannian manifold with ℝ.

From the fundamental Laplacian comparison theorem proved earlier by Eugenio Calabi, these functions are both superharmonic under the Ricci curvature assumption.

Then, the proof can be finished by using Bochner's formula to construct parallel vector fields, setting up the de Rham decomposition theorem.

[7] Proofs in various levels of generality were found by Jost Eschenburg, Gregory Galloway, and Richard Newman.