8.3 Bonnet-Myers theorem and its consequences

The Bonnet-Myers theorem connects positive Ricci curvature to an upper bound on a manifold's diameter. This powerful result shows how local curvature properties influence global geometry, leading to compactness and finite fundamental groups.

Consequences of the theorem extend to eigenvalue estimates, heat kernel bounds, and topological restrictions. These applications highlight the theorem's importance in geometric analysis and differential topology, bridging local and global aspects of Riemannian geometry.

Ricci Curvature and Diameter Bound

Ricci Curvature and Its Implications

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• Ricci curvature measures the average sectional curvature in different directions at a point on a Riemannian manifold
• Positive Ricci curvature indicates local convergence of geodesics, influencing global geometry
• Ricci curvature tensor $Ric(X,Y)$ defined as the trace of the linear map $R(X,\cdot,Y,\cdot)$, where $R$ is the Riemann curvature tensor
• Lower bounds on Ricci curvature lead to important geometric and topological consequences (sphere theorem, Bonnet-Myers theorem)
• Scalar curvature derived from Ricci curvature by taking its trace, providing a single number at each point

Diameter Bound and Compactness

• Diameter of a Riemannian manifold defined as the supremum of distances between any two points
• Bonnet-Myers theorem establishes a relation between positive Ricci curvature and upper bounds on diameter
• Diameter bound states that for an n-dimensional complete Riemannian manifold with $Ric \geq (n-1)K > 0$, the diameter is at most $\pi/\sqrt{K}$
• Compactness of the manifold follows from the diameter bound, as the manifold is both complete and has finite diameter
• Closed geodesics in manifolds with positive Ricci curvature have length bounded by $2\pi/\sqrt{K}$

Myers' Theorem and Its Applications

• Myers' theorem states that a complete Riemannian manifold with Ricci curvature bounded below by a positive constant is compact
• Provides a topological obstruction to the existence of complete metrics with positive Ricci curvature on non-compact manifolds
• Generalizes to the case where the Ricci curvature is bounded below by a function that grows sufficiently fast
• Applications include proving the finiteness of the fundamental group for manifolds with positive Ricci curvature
• Extends to study of Ricci solitons and other geometric structures in Riemannian geometry

Topological Consequences

Fundamental Group and Topology

• Fundamental group of a manifold with positive Ricci curvature is finite
• Finiteness follows from the compactness implied by Myers' theorem
• Universal cover of a manifold with positive Ricci curvature is compact, leading to finite fundamental group
• Positive Ricci curvature imposes restrictions on the possible topological types of manifolds
• Cheeger-Gromoll splitting theorem applies to manifolds with non-negative Ricci curvature, providing further topological information

Laplacian Eigenvalues and Spectral Properties

• First eigenvalue of the Laplacian on a compact Riemannian manifold relates to its geometry and topology
• Lichnerowicz estimate provides a lower bound for the first non-zero eigenvalue of the Laplacian in terms of Ricci curvature
• For an n-dimensional compact manifold with $Ric \geq (n-1)K > 0$, the first non-zero eigenvalue $\lambda_1 \geq nK$
• Spectral gap refers to the difference between the first two eigenvalues of the Laplacian
• Larger spectral gap indicates stronger mixing properties for heat flow and random walks on the manifold

Applications in Geometric Analysis

• Eigenvalue estimates used in the study of minimal surfaces and harmonic maps
• Cheeger's inequality relates the first non-zero eigenvalue to the isoperimetric constant of the manifold
• Heat kernel estimates derived from Ricci curvature bounds, with applications in probability theory
• Gradient estimates for harmonic functions on manifolds with Ricci curvature bounds
• Ricci flow, introduced by Hamilton, uses Ricci curvature to deform the metric and study the topology of 3-manifolds