Milnor conjecture (Ricci curvature)

The Milnor conjecture, in the context of Ricci curvature, is a statement concerning the relationship between the topology of a Riemannian manifold and its Ricci curvature. While there isn't a single, universally agreed-upon "Milnor conjecture" specifically about Ricci curvature in the same way there are Milnor conjectures in other areas of mathematics (like the Milnor conjecture on the knot concordance group), the term often refers to questions and results exploring bounds on topological invariants based on Ricci curvature restrictions.

These investigations frequently center around the following types of problems:

• Bounds on Betti numbers: Can the Ricci curvature of a manifold place restrictions on the number of its Betti numbers (which measure the "holes" of different dimensions in the manifold)? Results in this direction often involve establishing upper bounds on Betti numbers given lower bounds on Ricci curvature.

• Topological finiteness theorems: Can a lower bound on Ricci curvature, along with other geometric conditions like a diameter bound, guarantee that there are only finitely many diffeomorphism types of manifolds with that Ricci curvature? This is related to the Gromov's precompactness theorem but often with a focus on specific topological properties rather than general convergence.

• Relationship to scalar curvature: The Ricci curvature is a tensor, while the scalar curvature is a scalar function. Research sometimes investigates how bounds on Ricci curvature relate to bounds on scalar curvature and the resulting implications for topology.

It's crucial to note that there are no simple, universally applicable theorems directly analogous to the famous Milnor conjectures in other fields. The relationship between Ricci curvature and topology is complex and often depends heavily on the specific dimension and other geometric assumptions. Many open problems remain in this area of research. Specific results will depend on the context and the precise constraints placed on the Ricci curvature. Therefore, searching the literature for "Ricci curvature bounds and topology" or related phrases is necessary to understand the current state of research.