Rauch comparison theorem

Let ${\displaystyle M,{\widetilde {M}}}$ be Riemannian manifolds, on which are drawn unit speed geodesic segments ${\displaystyle \gamma$ :[0,T]\to M} and ${\displaystyle {\widetilde {\gamma }}:[0,T]\to {\widetilde {M}}}$. Assume that ${\displaystyle {\widetilde {\gamma }}(0)}$ has no conjugate points along ${\displaystyle {\widetilde {\gamma }}}$, and let ${\displaystyle J,{\widetilde {J}}}$ be two normal Jacobi fields along ${\displaystyle \gamma }$ and ${\displaystyle {\widetilde {\gamma }}}$ such that :

• ${\displaystyle J(0)=0}$ and ${\displaystyle {\widetilde {J}}(0)=0}$
• ${\displaystyle |D_{t}J(0)|=\left|{\widetilde {D}}_{t}{\widetilde {J}}(0)\right|}$.

If the sectional curvature of every 2-plane ${\displaystyle \Pi \subset T_{\gamma (t)}M}$containing ${\displaystyle {\dot {\gamma }}(t)}$ is less or equal than the sectional curvature of every 2-plane ${\displaystyle {\widetilde {\Pi }}\subset T_{{\tilde {\gamma }}(t)}{\widetilde {M}}}$ containing ${\displaystyle {\dot {\widetilde {\gamma }}}(t)}$, then ${\displaystyle |J(t)|\geq |{\widetilde {J}}(t)|}$ for all ${\displaystyle t\in [0,T]}$.

The theorem is formulated using Jacobi fields to measure the variation in geodesics. As the tangential part of a Jacobi field is independent of the geometry of the manifold, the theorem focuses on normal Jacobi fields, i.e. Jacobi fields which are orthogonal to the speed vector ${\displaystyle {\dot {\gamma }}(t)}$ of the geodesic for all time ${\displaystyle t}$. Up to reparametrization, every variation of geodesics induces a normal Jacobi field.

Jacobi fields are requested to vanish at time ${\displaystyle t=0}$ because the theorem measures the infinitesimal divergence (or convergence) of a family of geodesics issued from the same point ${\displaystyle \gamma (0)}$, and such a family induces a Jacobi field vanishing at ${\displaystyle t=0}$.

Under very similar conditions, it is also possible to compare the Hessian of the distance function to a given point.[1] It is also possible to compare the Laplacian of this function (which is the trace of the Hessian), with some additional condition on one of the two manifolds: it is then enough to have an inequality on the Ricci curvature (which is the trace of the curvature tensor).[1]

• Toponogov's theorem

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