Rauch comparison theorem

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 ${\displaystyle t\in [0,T]}$.

• Toponogov's theorem

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