8.4 Applications to manifolds with bounded curvature

Curvature bounds shape a manifold's geometry and topology. They influence volumes, injectivity radius, and even the existence of lines. These relationships provide powerful tools for understanding and classifying Riemannian manifolds.

Comparison theorems, like Bishop-Gromov, connect curvature bounds to volume ratios. The Cheeger-Gromoll splitting theorem and Gromov's compactness theorem reveal deep structural insights. These results highlight the profound impact of curvature on manifold properties.

Curvature Bounds and Volume Comparison

Sectional Curvature and Volume Relationships

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• Sectional curvature bounds define upper and lower limits for curvature in a Riemannian manifold
• Lower bounds on sectional curvature ($K \geq k$) imply the manifold curves at least as much as a space of constant curvature k
• Upper bounds on sectional curvature ($K \leq k$) indicate the manifold curves no more than a space of constant curvature k
• Bishop-Gromov volume comparison theorem relates volumes of geodesic balls in manifolds with bounded curvature to those in spaces of constant curvature
• For manifolds with $K \geq k$, the volume ratio $\frac{Vol(B(p,r))}{Vol(B_k(r))}$ decreases as r increases, where $B_k(r)$ is a ball in the space of constant curvature k
• This theorem provides powerful tools for estimating volumes and controlling geometry in manifolds with bounded curvature

Injectivity Radius and Geometric Implications

• Injectivity radius measures the largest radius for which the exponential map at a point remains injective
• In manifolds with bounded sectional curvature, the injectivity radius relates to the curvature bounds
• Lower bounds on sectional curvature often imply lower bounds on the injectivity radius
• Cheeger's lemma establishes a relationship between injectivity radius, volume, and diameter for manifolds with bounded curvature
• Applications include controlling the topology of manifolds with bounded curvature and positive injectivity radius (Cheeger finiteness theorem)

Splitting and Compactness Theorems

Cheeger-Gromoll Splitting Theorem and Its Consequences

• Cheeger-Gromoll splitting theorem applies to complete Riemannian manifolds with nonnegative Ricci curvature
• States that if such a manifold contains a line (a geodesic minimizing distance between any two of its points), it splits isometrically as a product $M = \mathbb{R} \times N$
• N represents another complete manifold with nonnegative Ricci curvature
• Implications for the structure of manifolds with nonnegative Ricci curvature (restricts possible topologies)
• Applications in studying the fundamental group and homotopy type of manifolds with nonnegative Ricci curvature

Gromov's Compactness Theorem and Precompactness

• Gromov's compactness theorem provides conditions under which a family of Riemannian manifolds is precompact in the Gromov-Hausdorff topology
• Precompactness means any sequence in the family has a convergent subsequence
• Applies to families of n-dimensional Riemannian manifolds with uniform bounds on diameter and sectional curvature
• Convergence occurs in the Gromov-Hausdorff sense, which measures how "close" two metric spaces are
• Precompactness allows for limiting arguments and proves existence of manifolds with specific properties
• Applications include studying Einstein manifolds and manifolds with special holonomy

Topological Invariants

Betti Numbers and Topological Structure

• Betti numbers represent topological invariants of a manifold, measuring the rank of homology groups
• First Betti number $b_1$ equals the rank of the first homology group, related to the number of "holes" in the manifold
• Higher Betti numbers $b_k$ measure higher-dimensional analogues of holes
• Bounds on curvature can impose restrictions on Betti numbers (Gromov's Betti number estimate)
• For manifolds with positive sectional curvature, all odd Betti numbers vanish except possibly in the middle dimension
• Applications include classifying manifolds and understanding their topological structure based on curvature bounds