Rauch comparison theorem

Remarks

The idea of the proof: for $0 < t \le a$, the quantity $\ip{\bar J'}{\bar J} / \abs{\bar J}^2$ at $t$ equals the index form of $\bar J$ on $[0, t]$ divided by $\abs{\bar J(t)}^2$. Since $\bar\gamma$ has no conjugate points, $\bar J$ minimizes the index form among fields with the same end values. Transplanting $J$ to $\bar\gamma$ by a linear isometry along the geodesics and using the curvature inequality shows that the logarithmic derivative of $\abs{J}$ is at least that of $\abs{\bar J}$. Integration from $0$, where the ratio $\abs{\bar J(t)} / \abs{J(t)}$ tends to $1$, gives the inequality.

Example: take $\bar M = \R^n$, where Jacobi fields with $\bar J(0) = 0$ are $\bar J(t) = t W$ for a constant vector $W$. If $M$ has sectional curvature $K \le 0$, then $\abs{J(t)} \ge t \abs{J'(0)}$ for normal Jacobi fields, so $J$ never vanishes for $t > 0$ and $M$ has no conjugate points. This is the key step of the Cartan–Hadamard theorem.

Hypothesis 1 is needed: on the unit sphere, normal Jacobi fields $\sin(t) E(t)$ vanish at $t = \pi$, and comparisons with the sphere hold only up to that first conjugate point.

Going deeper

For a student

Rauch's theorem is the infinitesimal tool behind much of comparison geometry. Comparing with the space forms of constant curvature gives explicit bounds on Jacobi fields, on conjugate points and on the differential of the exponential map. It is used in the proofs of the Cartan–Hadamard theorem, of Toponogov's triangle comparison theorem and of the sphere theorem.