Splitting and Soul Theorems

The compactness theorems of 9B.4 Convergence of Manifolds produce limits; this chapter says what limits with nonnegative curvature can look like. The answer comes from three structure theorems. The splitting theorem: nonnegative Ricci curvature plus a single straight line forces the manifold to be a product $\mathbb{R}\times N$. The soul theorem: a complete noncompact manifold with nonnegative sectional curvature is a vector bundle over a compact totally geodesic submanifold, its soul. And the asymptotic cone: such a manifold, viewed from very far away, looks like a cone.

These theorems have few direct applications outside geometry, and this chapter does not invent any. Their use is internal. In Perelman's classification of the singularity models of the three-dimensional Ricci flow (12B.2 The Structure of κ-Solutions), a line in a limit forces a splitting, which produces the round cylinder $S^2\times\mathbb{R}$, the neck. The soul theorem says that every other noncompact model is diffeomorphic to $\mathbb{R}^3$, a cap. Necks and caps are the two pieces of the canonical neighbourhood theorem.

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

• define rays, lines and Busemann functions, and compute them in model spaces;
• prove the splitting theorem from Laplacian comparison, the maximum principle and Bochner's formula;
• state Toponogov's triangle comparison and check it on the sphere;
• state the soul theorem and Perelman's solution of the soul conjecture, and find souls of examples;
• describe asymptotic cones, and say what these results mean for the neck and the cap.

5.1Lines and Busemann functions

A ray is a unit-speed geodesic $\gamma : [0, \infty) \to M$ that minimises between any two of its points; a line is such a geodesic defined on all of $\mathbb{R}$. Every complete noncompact manifold has a ray from each point (a limit of minimising segments to points going to infinity). Lines are much rarer. A paraboloid has rays but no line, while a cylinder has many lines.

The Busemann function of a ray $\gamma$ is

The limit exists because the expression is nonincreasing in $t$ (triangle inequality) and bounded below by $-d(x, \gamma(0))$. It is $1$-Lipschitz. It measures "how far behind" $x$ is in a race to infinity along $\gamma$. In $\mathbb{R}^n$, for $\gamma(t) = tv$, $b_\gamma(x) = -\langle x, v\rangle$ (Exercise 5.4): its level sets are the hyperplanes orthogonal to $v$. In the hyperbolic plane they are horocycles.

5.2The splitting theorem

Applying the theorem repeatedly, a complete manifold with $\operatorname{Ric} \geq 0$ is isometric to $\mathbb{R}^k\times N$, where $N$ contains no line. For a closed manifold with $\operatorname{Ric} \geq 0$, the universal cover is $\mathbb{R}^k\times N$ with $N$ compact (Cheeger–Gromoll), which constrains the fundamental group: it has a finite normal subgroup whose quotient contains $\mathbb{Z}^k$ as a subgroup of finite index.

5.3Triangles: Toponogov's theorem

Sectional curvature bounds control triangles. A geodesic triangle has three minimising geodesic sides; its comparison triangle in the model space of curvature $k$ has the same side lengths.

With $k = 0$: triangles in a manifold of nonnegative curvature are "fatter" than Euclidean ones, and their angle sums are at least $\pi$, as on the sphere (Figure 5.2, Exercise 5.7). Victor Toponogov proved the theorem in 1959. It works with distances alone, so it makes sense on singular limits too. Alexandrov spaces, metric spaces in which Toponogov's conclusion holds, are the natural class of Gromov–Hausdorff limits of manifolds with $K \geq k$.

5.4Souls

The soul is found by a convexity argument. Busemann functions are concave when $K \geq 0$ (by Toponogov). Their superlevel sets form an exhausting family of compact totally convex sets, and one shrinks such a set until no interior is left, repeating the process on its boundary structure if necessary. Detlef Gromoll and Wolfgang Meyer had proved in 1969 that $K > 0$ everywhere forces $M$ to be diffeomorphic to $\mathbb{R}^n$. Cheeger and Gromoll conjectured that $K \geq 0$ everywhere and $K > 0$ at a single point should be enough. Perelman proved this soul conjecture in 1994: then the soul is a point, and $M$ is diffeomorphic to $\mathbb{R}^n$.

Examples (Exercise 5.8): a paraboloid has a point as its soul and is diffeomorphic to $\mathbb{R}^2$; the cylinder $S^1\times\mathbb{R}$ has any circle $S^1\times\{t\}$ as a soul; the flat Möbius band has its central circle; $S^2\times\mathbb{R}$ has $S^2\times\{t\}$; a capped half-cylinder, positively curved on its cap, has a point.

5.5The view from infinity

If $K \geq 0$, the blow-downs $(M, \lambda^{-2}g, p)$, $\lambda \to \infty$, converge in the pointed Gromov–Hausdorff sense to a metric cone $C(X)$ over a compact space $X$, the asymptotic cone of $M$. The proof uses the monotonicity of angles that Toponogov's theorem provides. For a paraboloid, the asymptotic cone is a ray (Figure 5.3); for $\mathbb{R}^n$ it is $\mathbb{R}^n$; for a cone, the cone itself. The asymptotic volume ratio of 9B.2 Volume Comparison is positive exactly when the asymptotic cone is $n$-dimensional.

5.6History

Herbert Busemann introduced his functions in The Geometry of Geodesics (1955). Toponogov proved his comparison theorem in 1959, and the splitting theorem for $K \geq 0$ in 1964. Jeff Cheeger and Detlef Gromoll proved the splitting theorem for $\operatorname{Ric} \geq 0$ in 1971 and the soul theorem in 1972; Gromoll and Meyer's theorem on $K > 0$ dates from 1969. Jost-Hinrich Eschenburg and Ernst Heintze gave the short proof of splitting followed above in 1984. Perelman's proof of the soul conjecture appeared in the Journal of Differential Geometry in 1994.

5.7Exercises

Solution

(a) $|x - tv| - t = \sqrt{t^2 - 2t\langle x, v\rangle + |x|^2} - t \to -\langle x, v\rangle$. (b) For $w = ie^t$, $t \to \infty$: $|z - w|^2 \sim e^{2t}$, so $\cosh d \sim \frac{e^t}{2y}$ ($y = \operatorname{Im}z$) and $d = t - \log y + o(1)$, giving $b^+ = -\log y$. For $w = ie^{-t}$: $|z - w|^2 \to |z|^2$, so $\cosh d \sim \frac{|z|^2e^t}{2y}$, $d = t + \log\frac{|z|^2}{y} + o(1)$, giving $b^- = 2\log|z| - \log y$. (c) $|z| \geq y$ with equality on the imaginary axis. The hyperbolic plane has $\operatorname{Ric} = -g < 0$, so the theorem does not apply; indeed $b^\pm$ are not superharmonic there.

Solution

$\frac12\Delta|\nabla b|^2 = 0 = |\nabla^2b|^2 + \langle\nabla b, \nabla\Delta b\rangle + \operatorname{Ric}(\nabla b, \nabla b) = |\nabla^2b|^2 + \operatorname{Ric}(\nabla b, \nabla b)$, a sum of nonnegative terms. If $\nabla X = 0$, then $(\mathcal L_Xg)(Y, Z) = \langle\nabla_YX, Z\rangle + \langle Y, \nabla_ZX\rangle = 0$.

Solution

$\operatorname{Ric} = g_{S^2}\oplus0 \geq 0$ (9A.5 Computing Curvature). $d((x, s), (x, t)) = |s - t|$ because the distance in a product is $\sqrt{d_{S^2}^2 + |s - t|^2}$. $S^3$ is compact, so it has no geodesic minimising on all of $\mathbb{R}$. On the paraboloid ($K > 0$, $\operatorname{Ric} \geq 0$), a line would split it as $\mathbb{R}\times N$, forcing $K = 0$ in some plane at every point, which is false.

Solution

$\cos\alpha = \frac{\cos a - \cos^2a}{\sin^2a} = \frac{\cos a(1 - \cos a)}{(1 - \cos a)(1 + \cos a)} = \frac{\cos a}{1 + \cos a}$. This is $< \frac12$ iff $2\cos a < 1 + \cos a$ iff $\cos a < 1$, true for $a > 0$; and $\alpha$ is defined while $\cos a > -\frac12$. For $a = \frac{\pi}{2}$: $\cos\alpha = 0$, three right angles, sum $\frac{3\pi}{2}$ (the octant of 9A.2 Connections).

Solution

(a) The vertex; $M \cong \mathbb{R}^2$, the normal bundle of a point. (b) Any circle $S^1\times\{t\}$; $M$ is its trivial line bundle. (c) $S^2\times\{t\}$; $M = S^2\times\mathbb{R}$, the trivial line bundle. (d) $K > 0$ on the cap, so by Perelman's theorem the soul is a point (the tip of the cap, by symmetry); $M \cong \mathbb{R}^2$.

Solution

(a) $K \geq 0$ implies $\operatorname{Ric} \geq 0$, so the splitting theorem applies; the curvature of $N$ is a sectional curvature of $M$, so $K_N \geq 0$, and $N$ is orientable because $M$ is. (b) $\int_NK\,dA = 2\pi\chi(N) > 0$, so $\chi(N) = 2$ and $N$ is a sphere. (c) A split $\kappa$-solution is $\mathbb{R}\times N$ with $N$ an ancient solution of the two-dimensional Ricci flow with positive curvature on $S^2$, defined for all negative times, and noncollapsed. On $S^2$, the only such solutions that remain noncollapsed are the round shrinking spheres (the classification of ancient solutions on $S^2$, 11B.2 Ancient Solutions and the Harnack Inequality), so the factor is round.