# Are there references that talk about the connection between curvature and compactness?

The Killing-Hopf theorem says that if $$M$$ is a complete connected Riemannian manifold of constant sectional curvature $$K$$, its universal cover is one of the following:

• if $$K > 0$$, it is the sphere;
• if $$K = 0$$, it is the Euclidean space;
• if $$K < 0$$, it is the hyperbolic space.

Of these possibilities, only the first one implies a compact topology.

Are there any texts or references that delve into the connection between compactness and curvature more deeply? Is there anything that classifies manifolds in the spirit of the Killing-Hopf theorem for more general cases?

• Well, there's two theorems of differential geometry that might be relevant to your question. One, the Cartan-Hadamard theorem, says every simply connected, geodesically complete Riemannian manifold whose sectional curvatures are all $\le 0$ is homeomorphic to Euclidean space (and hence is noncompact). The other says that every geodesically complete Riemannian manifold whose sectional curvatures are all $\ge c$ for some $c>0$ is compact (and hence has finite fundamental group, because its universal cover satisfies the same hypotheses and hence is also compact). Is that what you had in mind? – Lee Mosher Jul 9 at 1:21
• To be compact, a space must be closed and bounded. The sphere is the only one of the three above that is closed and bounded. Not all manifolds are of constant curvature. A torus is a compact manifold that has regions that are positive curved and regions that are negatively curved. – Doug M Jul 9 at 1:24
• @LeeMosher Yeah, those are exactly the kinds of things I had in mind. – Maximal Ideal Jul 9 at 1:53
• Okay, I've turned my comments into an answer. – Lee Mosher Jul 9 at 2:01

There is an entire subfield of geometry that studies relations between curvature and topology, including compactness, it is called geometry in the large. It had its prime in 1950-60s, the geometrization conjecture of Thurston is an outgrowth of it for $$3$$-manifolds. The classical book on this subject is Gromoll, D.; Klingenberg, W.; Meyer, W. Riemannsche Geometrie im Großen (Riemannian Geometry in the Large). From MathSciNet:

Chapter 6 presents a number of important comparison theorems, including those of Morse-Schoenberg and Rauch, and culminates with the angle comparison theorem of Toponogov. With Chapter 7 we enter upon the main subject: the relation between curvature and topological structure. The exposition begins with some classical results, such as the Hadamard-Cartan theorem for spaces with nonpositive sectional curvature and the theorem of Myers for spaces of positive curvature. This is followed by a closely related result of Toponogov, a discussion of orientability, the Synge lemma and corollary (a compact connected orientable Riemannian manifold of even dimension having positive sectional curvature is simply connected). The development culminates and concludes with the Rauch-Klingenberg sphere theorem, proved via Morse theory and the angle comparison theorem of Toponogov.

Unfortunately, I do not think it has been translated into English. If German is not your thing, the material can be found in Cheeger-Ebin Comparison Theorems in Riemannian Geometry (freely available).

The last four chapters (6–9) form the "core'' of the authors' study. Chapter 6 deals with the sphere theorem (Rauch, Klingenberg, Berger), the maximal diameter theorem (Toponogov), and the minimal diameter theorem (Berger). Chapter 7 studies the differential sphere theorem and closes with a discussion of general problems concerning compact manifolds of positive curvature. The concept of a soul S, a compact totally geodesic and totally convex submanifold, is central to the study of complete (non-compact) manifolds M of nonnegative curvature treated in Chapter 8... Chapter 9 studies compact manifolds of non-positive curvature giving some results on the fundamental group.

See also Geometry of Spaces of Constant Curvature by Alekseevskij et al. for a more specialized topic.

There are two relevant theorems of differential geometry.

One is the Cartan-Hadamard theorem, which says that if $$M$$ is a geodesically complete, simply connected Riemannian $$m$$-manifold such that all sectional curvatures of $$M$$ are $$\le 0$$, then $$M$$ is diffeomorphic to Euclidean space $$\mathbb R^m$$, and so in particular $$M$$ is noncompact. One application is that every compact Riemannian manifold whose sectional curvatures are $$\le 0$$ has infinite fundamental group, because its universal cover satisfies the hypotheses of the Cartan-Hadamard theorem and hence is not compact.

The other is the Bonnet-Myer theorem. A weak form of this theorem (due to Bonnet) says that if $$M$$ is a geodesically complete Riemannian $$m$$-manifold such that all sectional curvatures of $$M$$ are $$> k$$ where $$k$$ is some positive constant, then $$M$$ is compact. It follows that the fundamental group of $$M$$ is finite, because the same hypotheses hold for the universal cover of $$M$$ which is therefore compact.