Cheeger–Gromoll splitting theorem

Remarks

The idea of the proof: let $b^{+}$ and $b^{-}$ be the Busemann functions of the two rays $t \mapsto \gamma(t)$ and $t \mapsto \gamma(-t)$, $t \ge 0$. The Laplacian comparison theorem gives $\Delta b^{\pm} \le 0$ in a weak sense. The triangle inequality gives $b^{+} + b^{-} \ge 0$, with equality on $\gamma$, so the strong maximum principle forces $b^{+} + b^{-} = 0$ and $b^{+}$ is harmonic and smooth. The Bochner formula applied to $b^{+}$, with $\abs{\nabla b^{+}} = 1$, gives that its Hessian vanishes, so $\nabla b^{+}$ is a parallel vector field and $M$ splits along its flow.

Example: the flat cylinder $S^1 \times \R$ contains the lines $\set{\theta} \times \R$ and splits with $N = S^1$.

The curvature hypothesis is needed: the hyperbolic plane contains lines but is not a product with a line. The line is needed too: the paraboloid $z = x^2 + y^2$ has positive curvature and contains no line.

Going deeper

For a student

Toponogov proved the theorem earlier under the stronger hypothesis of nonnegative sectional curvature; Cheeger and Gromoll proved the Ricci version in 1971. A main consequence: if $M$ is compact with $\operatorname{Ric} \ge 0$, then its universal cover is isometric to $N \times \R^k$ with $N$ compact. It follows that the fundamental group of such $M$ contains a free abelian subgroup of finite index. Versions of the theorem for limit spaces are central in the work of Cheeger and Colding.