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.