Splitting theorem

The splitting theorem is a classical theorem in Riemannian geometry. It states that if a complete Riemannian manifold M with Ricci curvature

${\displaystyle {\rm {Ric}}(M)\geq 0}$

has a straight line, i.e., a geodesic γ such that

${\displaystyle d(\gamma (u),\gamma (v))=|u-v|}$

for all

${\displaystyle u,v\in \mathbb {R} ,}$

then it is isometric to a product space

${\displaystyle \mathbb {R} \times L,}$

where ${\displaystyle L}$ is a Riemannian manifold with

${\displaystyle {\rm {Ric}}(L)\geq 0.}$