Jacobi Fields and Curvature versus Topology

Curvature is local: it is computed at a point from second derivatives of the metric. Topology is global. This chapter shows how to get from one to the other, and the bridge is the behaviour of families of geodesics. Nearby geodesics leaving a point separate at a rate governed by curvature, through the Jacobi equation, a linear ODE along each geodesic. Positive curvature makes them reconverge, so geodesics stop minimising after a bounded length. Negative curvature keeps them apart forever.

Two theorems turn this into topology. Bonnet–Myers: a complete manifold with Ricci curvature bounded below by a positive constant is compact, with finite fundamental group. Cartan–Hadamard: a complete, simply connected manifold with nonpositive sectional curvature is diffeomorphic to $\mathbb{R}^n$. The Ricci flow's whole strategy is to make curvature positive, or to cut the manifold into pieces where it is, so that theorems of this kind identify the topology. Hamilton's 1982 theorem starts where Myers stops.

By the end of this chapter you will be able to:

• derive the first and second variation formulas for length, and write the index form;
• define Jacobi fields, solve the Jacobi equation in constant curvature, and relate Jacobi fields to the exponential map;
• define conjugate points and explain why geodesics stop minimising past them;
• prove the Bonnet–Myers theorem, and state Synge's and the Cartan–Hadamard theorems;
• derive the normal-coordinate expansion of the metric from Jacobi fields.

7.1Gravitational lensing

7.2Variations of length

A variation of a curve $\gamma : [0, L] \to M$ is a smooth family $\Gamma(s, t)$ with $\Gamma(0, t) = \gamma(t)$; its variation field is $V(t) = \partial_s\Gamma(0, t)$. It is proper if the endpoints are fixed. Write $T = \partial_t\Gamma$, $S = \partial_s\Gamma$, and recall the symmetry lemma $D_sT = D_tS$ (from torsion-freeness, used in the Gauss lemma of 9A.3 Geodesics and the Exponential Map).

So a unit-speed curve is a critical point of length for all proper variations exactly when $D_t\gamma' = 0$: geodesics are the critical points of length. Whether a geodesic minimises is a second-order question.

Curvature enters with a minus sign. Positive sectional curvature along a geodesic makes some variations shorten it, and a long enough geodesic on a positively curved manifold can always be shortened.

7.3Jacobi fields

A Jacobi field along a geodesic $\gamma$ is a vector field $J$ along it satisfying the Jacobi equation

It is a linear second-order ODE, so a Jacobi field is determined by $J(0)$ and $D_tJ(0)$, and they form a $2n$-dimensional space. The variation field of any variation through geodesics is a Jacobi field (Exercise 7.6), and conversely. In particular, for a geodesic $\gamma(t) = \exp_p(tv)$ and $w \in T_pM$,

is the Jacobi field with $J(0) = 0$ and $D_tJ(0) = w$, the variation field of the fan of geodesics $s \mapsto \exp_p(t(v + sw))$. Jacobi fields measure how fast nearby geodesics from $p$ spread apart.

Constant curvature. If $K \equiv k$, then $R(J, \gamma')\gamma' = k\,J$ for unit-speed $\gamma$ and $J \perp \gamma'$, and the Jacobi fields vanishing at $0$ and orthogonal to $\gamma$ are

with $E$ parallel (Exercise 7.7, Figure 7.2). On the sphere they vanish again at $t = \frac{\pi}{\sqrt k}$; in the plane they grow linearly; in hyperbolic space exponentially. These are the functions $\operatorname{sn}_k$ of the polar metrics of 9A.1 Riemannian Metrics and Model Spaces, $dr^2 + \operatorname{sn}_k(r)^2g_{S^{n-1}}$, for the same reason.

Conjugate points. $\gamma(t_0)$ is conjugate to $\gamma(0)$ along $\gamma$ if a nonzero Jacobi field vanishes at both. Equivalently, $d(\exp_p)$ is singular at $t_0v$: the fan of geodesics from $p$ focuses at $\gamma(t_0)$. A geodesic does not minimise length past its first conjugate point: the index form then has a negative direction (Jacobi's theorem, Lee, chapter 10). Lensing is this: the source and the observer are conjugate along the rays.

This is the expansion used in 9A.3 Geodesics and the Exponential Map and 9A.4 Curvature and What It Means; for unit $w \perp v$ it gives circumference $2\pi r(1 - \frac K6r^2 + \dots)$ of small geodesic circles (Exercise 7.11).

7.4Positive curvature: Bonnet–Myers

The unit sphere shows the diameter bound is sharp. Some consequences:

• $S^2\times S^1$, $T^3$ and every manifold with infinite fundamental group carry no metric with $\operatorname{Ric} > 0$, although $S^2\times S^1$ carries one with $\operatorname{Ric} \geq 0$ (9A.5 Computing Curvature).
• A closed 3-manifold with $\operatorname{Ric} > 0$ has finite fundamental group. Hamilton's theorem (11A.6 Hamilton’s 1982 Theorem) goes much further: such a manifold is diffeomorphic to a quotient $S^3/\Gamma$ of the round sphere, because the normalised Ricci flow deforms the metric to one of constant positive curvature.

Synge's theorem (1936) uses the same second variation idea along a shortest closed geodesic: a compact, even-dimensional manifold with positive sectional curvature is simply connected if it is orientable, and has fundamental group $\mathbb{Z}/2$ if it is not. So $\mathbb{RP}^2\times\mathbb{RP}^2$, compact and four-dimensional with fundamental group of order $4$, has no metric of positive sectional curvature.

7.5Nonpositive curvature: Cartan–Hadamard

The key step is that there are no conjugate points: if $K \leq 0$ and $J(0) = 0$, then $\frac{d^2}{dt^2}|J|^2 = 2|D_tJ|^2 - 2\operatorname{Rm}(J, \gamma', \gamma', J) \geq 0$, so $|J|^2$ is convex and a nonzero Jacobi field cannot vanish twice (Exercise 7.8). So $\exp_p$ is a local diffeomorphism everywhere, and completeness makes it a covering map (Lee, chapter 12). Consequences:

• Manifolds with $K \leq 0$ are aspherical: all higher homotopy groups vanish. Flat tori, hyperbolic surfaces and closed hyperbolic 3-manifolds are examples, and the hyperbolic pieces of the geometrization theorem are of this kind (10A.6 Hyperbolic Three-Manifolds).
• $S^n$, $n \geq 2$, has no metric with $K \leq 0$: it is compact and simply connected, so it would be diffeomorphic to $\mathbb{R}^n$.

7.6History

Carl Jacobi studied the second variation of geodesics on surfaces in 1837, and Pierre Ossian Bonnet proved the diameter bound for surfaces with Gauss curvature bounded below in 1855. Sumner Byron Myers extended it to Ricci curvature in 1941. John Lighton Synge proved his theorem in 1936. Jacques Hadamard showed in 1898 that surfaces of negative curvature have no conjugate points, and Élie Cartan proved the general covering theorem in 1928. Marston Morse developed the index theory of geodesics in the 1930s. The first gravitational lens was found in 1979.

7.7Exercises

Solution

$0 = D_sD_tT = D_tD_sT + R(S, T)T = D_tD_tS + R(S, T)T$. At $s = 0$: $D_t^2V + R(V, \gamma')\gamma' = 0$.

Solution

$R(J, \gamma')\gamma' = k(\langle\gamma', \gamma'\rangle J - \langle J, \gamma'\rangle\gamma') = kJ$. For $J = fE$, $D_t^2J = f''E$, so the equation is $(f'' + kf)E = 0$. With $f(0) = 0$, $f = c\operatorname{sn}_k$; for $k > 0$ it vanishes again first at $t = \pi/\sqrt k$, the first conjugate point; for $k \leq 0$ never.

Solution

$\frac{d}{dt}|J|^2 = 2\langle D_tJ, J\rangle$ and $\frac{d^2}{dt^2}|J|^2 = 2|D_tJ|^2 + 2\langle D_t^2J, J\rangle = 2|D_tJ|^2 - 2\langle R(J, \gamma')\gamma', J\rangle$, and $\langle R(J, \gamma')\gamma', J\rangle = \operatorname{Rm}(J, \gamma', \gamma', J) \leq 0$. So $|J|^2$ is convex, with $|J|^2(0) = 0$ and $|J|^2 = t^2|D_tJ(0)|^2 + O(t^3) > 0$ for small $t > 0$. A convex function that is zero at $0$ and positive just after cannot return to zero.

Solution

(a) $\operatorname{Ric} = (n - 1)g$, $k = 1$, and antipodal points are at distance $\pi$. (b) Its curvature tends to $0$ at infinity, so there is no positive lower bound $k$. (c) On a compact manifold the continuous function $\min_{|v| = 1}\operatorname{Ric}(v, v)$ attains a positive minimum, which gives a $k > 0$.

Solution

Take the line of intersection of the two planes as the reference. The tilted satellite's height above the first plane is $a\sin\delta\sin(nt) \approx a\delta\sin nt$, which solves $\ddot z + n^2z = 0$. It vanishes when $nt$ is a multiple of $\pi$, so every half orbit: every $45$ minutes.

Solution

$\partial_\theta C_r = d(\exp_p)_{ru_\theta}(ru_\theta')$, which is $J_\theta(r)$ for the Jacobi field $d(\exp_p)_{tu_\theta}(tu_\theta')$. From the proof of the proposition, $|J(t)|^2 = t^2 - \frac{t^4}{3}\operatorname{Rm}(u_\theta', u_\theta, u_\theta, u_\theta') + O(t^5) = t^2 - \frac{t^4}{3}K(\Pi) + O(t^5)$, since $u_\theta, u_\theta'$ is an orthonormal basis of $\Pi$. So $|J(r)| = r(1 - \frac{K}{6}r^2 + O(r^3))$, and integrating over $\theta \in [0, 2\pi]$ gives $L(C_r) = 2\pi r(1 - \frac{K(\Pi)}{6}r^2 + O(r^3))$.