Rauch comparison theorem

In Riemannian geometry, the Rauch comparison theorem, named after Harry Rauch who proved it in 1951, is a fundamental result which relates the sectional curvature of a Riemannian manifold to the rate at which geodesics spread apart. Intuitively, it states that for positive curvature, geodesics tend to converge, while for negative curvature, geodesics tend to spread. This theorem is formulated using Jacobi fields to measure the variation in geodesics.

Statement

Let ${\displaystyle M,{\widetilde {M}}}$ be Riemannian manifolds, let ${\displaystyle \gamma :[0,T]\to M}$ and ${\displaystyle {\widetilde {\gamma }}:[0,T]\to {\widetilde {M}}}$ be unit speed geodesic segments such that ${\displaystyle {\widetilde {\gamma }}(0)}$ has no conjugate points along ${\displaystyle {\widetilde {\gamma }}}$, and let ${\displaystyle J,{\widetilde {J}}}$ be normal Jacobi fields along ${\displaystyle \gamma }$ and ${\displaystyle {\widetilde {\gamma }}}$ such that ${\displaystyle J(0)={\widetilde {J}}(0)=0}$ and ${\displaystyle |D_{t}J(0)|=|{\widetilde {D}}_{t}{\widetilde {J}}(0)|}$. Suppose that the sectional curvatures of ${\displaystyle M}$ and ${\displaystyle {\widetilde {M}}}$ satisfy ${\displaystyle K(\Pi )\leq {\widetilde {K}}({\widetilde {\Pi }})}$ whenever ${\displaystyle \Pi \subset T_{\gamma (t)}M}$ is a 2-plane containing ${\displaystyle {\dot {\gamma }}(t)}$ and ${\displaystyle {\widetilde {\Pi }}\subset T_{{\tilde {\gamma }}(t)}{\widetilde {M}}}$ is a 2-plane containing ${\displaystyle {\dot {\widetilde {\gamma }}}(t)}$. Then ${\displaystyle |J(t)|\geq |{\widetilde {J}}(t)|}$ for all ${\displaystyle t\in [0,T]}$.

See also

• Toponogov's theorem

References

• do Carmo, M.P. Riemannian Geometry, Birkhäuser, 1992.
• Lee, J. M., Riemannian Manifolds: An Introduction to Curvature, Springer, 1997.