Proof of the Rauch Comparison Theorem

The Rauch comparison theorem is the fundamental result of comparison geometry, relating the growth of Jacobi fields on a manifold to those on a model space via sectional curvature bounds.

Statement

Theorem9.2Rauch comparison theorem

Let $\gamma: [0, T] \to M$ and $\bar\gamma: [0, T] \to \bar M$ be unit-speed geodesics with no conjugate points on $(0, T]$ . Suppose the sectional curvatures satisfy

K_M(\sigma) \leq K_{\bar M}(\bar\sigma)

for all 2-planes $\sigma$ containing $\gamma'$ and $\bar\sigma$ containing $\bar\gamma'$ . Let $J$ and $\bar J$ be Jacobi fields along $\gamma$ and $\bar\gamma$ respectively, with $J(0) = \bar J(0) = 0$ , $|J'(0)| = |\bar J'(0)|$ , and both orthogonal to the geodesics. Then

|J(t)| \geq |\bar J(t)| \quad \text{for all } t \in [0, T].

Proof

Step 1: Reduction to a scalar comparison. Set $f(t) = |J(t)|$ and $\bar f(t) = |\bar J(t)|$ . Both vanish at $t = 0$ and are smooth for $t > 0$ (away from zeros). We want to show $f(t) \geq \bar f(t)$ .

Step 2: Logarithmic derivative. For $t > 0$ , compute

\frac{d}{dt}\log f(t) = \frac{f'(t)}{f(t)} = \frac{g(J', J)}{|J|^2},

where $J' = \nabla_{\gamma'} J$ . The key quantity is the index form ratio:

u(t) = \frac{f'(t)}{f(t)}, \quad \bar u(t) = \frac{\bar f'(t)}{\bar f(t)}.

Step 3: Riccati comparison. Using the Jacobi equation $J'' + R(J, \gamma')\gamma' = 0$ and the definition of sectional curvature, one shows that $u$ satisfies the Riccati-type inequality:

u'(t) \leq -u(t)^2 - K_M(t),

where $K_M(t) = K(\mathrm{span}(J(t), \gamma'(t)))$ (with appropriate averaging if $J$ is not simple). Similarly, $\bar u'(t) = -\bar u(t)^2 - K_{\bar M}(t)$ .

Step 4: Comparison. Since $K_M \leq K_{\bar M}$ :

u' - \bar u' \leq -(u^2 - \bar u^2) - (K_M - K_{\bar M}) \leq -(u + \bar u)(u - \bar u).

Setting $w = u - \bar u$ : $w' \leq -(u + \bar u)w$ .

Step 5: Initial condition. As $t \to 0^+$ : $f(t) \sim t|J'(0)|$ , $\bar f(t) \sim t|\bar J'(0)|$ , so $u(t) \sim 1/t \sim \bar u(t)$ , giving $w(t) \to 0$ as $t \to 0$ .

The differential inequality $w' \leq -(u + \bar u)w$ with $w(0^+) = 0$ implies, by a comparison argument, that $w(t) \leq 0$ for all $t$ where both $f$ and $\bar f$ are positive. (If $w$ were to become positive at some first time $t_1$ , then $w' \leq 0$ at $t_1$ , contradicting $w$ becoming positive.)

Step 6: Conclusion. $u(t) \leq \bar u(t)$ means $(\log f)' \leq (\log \bar f)'$ , so $\log(f/\bar f)$ is non-increasing. Since $\lim_{t \to 0} f(t)/\bar f(t) = |J'(0)|/|\bar J'(0)| = 1$ , we get $f(t)/\bar f(t) \geq 1$ , i.e., $|J(t)| \geq |\bar J(t)|$ . $\blacksquare$

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Consequences

ExampleImmediate corollaries

Conjugate point comparison: If $K_M \leq \kappa$ , then conjugate points along any geodesic in $M$ occur no earlier than $\pi/\sqrt{\kappa}$ (when $\kappa > 0$ ). If $K_M \geq \kappa > 0$ , they occur no later.
Injectivity radius bound: $K \leq \kappa$ implies $\mathrm{inj}(M) \geq \mathrm{inj}(M^n_\kappa)$ .
Distance comparison: Combined with the exponential map, Rauch comparison gives global distance estimates (Toponogov's theorem).

RemarkThe Riccati equation approach

The proof uses the Riccati equation for the logarithmic derivative of the Jacobi field norm. This technique -- reducing the tensor Jacobi equation to a scalar Riccati comparison -- is a central method in comparison geometry, also used in the proof of the Bishop-Gromov volume comparison and the Laplacian comparison theorem.