🌀Riemannian Geometry Unit 8 – Comparison and Bonnet-Myers Theorems

Key Concepts and Definitions

• Riemannian manifold a smooth manifold equipped with a Riemannian metric, which is a positive-definite inner product on each tangent space
• Sectional curvature a measure of the curvature of a Riemannian manifold, defined as the Gaussian curvature of a geodesic surface formed by two orthonormal tangent vectors
  • Positive sectional curvature indicates that geodesics converge, while negative sectional curvature implies geodesics diverge
• Geodesic a locally length-minimizing curve on a Riemannian manifold, generalizing the concept of straight lines in Euclidean space
• Conjugate points pairs of points along a geodesic where the geodesic fails to minimize length locally
• Diameter the maximum distance between any two points on a Riemannian manifold, defined using the Riemannian distance function
• Injectivity radius the largest radius for which the exponential map is a diffeomorphism at every point of the manifold
• Jacobi field a vector field along a geodesic that measures the variation of the geodesic under small perturbations of its initial conditions

Historical Context and Development

• Bernhard Riemann introduced the concept of Riemannian geometry in his 1854 habilitation lecture, generalizing Gauss's theory of surfaces to higher dimensions
• Elwin Bruno Christoffel developed the fundamental tools of Riemannian geometry, including the Christoffel symbols and the Riemann curvature tensor, in the 1860s
• Pierre Ossian Bonnet proved the Bonnet-Myers theorem in 1855, establishing a fundamental link between curvature and topology in Riemannian geometry
• Hermann von Helmholtz and Sophus Lie made significant contributions to the development of comparison theorems in the late 19th century
• Harry Rauch proved the Rauch comparison theorem in 1951, which compares Jacobi fields on manifolds with different curvature bounds
• Jeff Cheeger and Detlef Gromoll proved the soul theorem in 1972, a powerful generalization of the Bonnet-Myers theorem for complete non-compact manifolds
• Mikhael Gromov introduced the notion of Gromov-Hausdorff convergence in the 1980s, providing a framework for studying the limit behavior of sequences of Riemannian manifolds

Comparison Theorems: Fundamentals

• Comparison theorems relate geometric properties of Riemannian manifolds with different curvature bounds
• Rauch comparison theorem compares Jacobi fields on manifolds with sectional curvature bounded above or below by a constant
  • If $M$ has sectional curvature $\leq \kappa$ and $\tilde{M}$ has constant sectional curvature $\kappa$, then Jacobi fields on $M$ grow faster than corresponding Jacobi fields on $\tilde{M}$
• Toponogov comparison theorem compares the angle and distance relationships in geodesic triangles on manifolds with sectional curvature bounded below
  • If $M$ has sectional curvature $\geq \kappa$, then geodesic triangles in $M$ are "thinner" than corresponding triangles in the space of constant curvature $\kappa$
• Hessian comparison theorem relates the Hessian of the distance function on a manifold to the corresponding Hessian in a space of constant curvature
• Volume comparison theorem compares the volume of balls in a manifold with sectional curvature bounded above or below to the volume of balls in a space of constant curvature
• Laplacian comparison theorem compares the Laplacian of the distance function on a manifold with the Laplacian in a space of constant curvature

Bonnet-Myers Theorem: Statement and Significance

• Bonnet-Myers theorem states that a complete Riemannian manifold with Ricci curvature bounded below by a positive constant is compact and has a finite fundamental group
  • Specifically, if $M$ is a complete Riemannian manifold with $\text{Ric} \geq (n-1)\kappa > 0$, then $M$ is compact with diameter $\leq \frac{\pi}{\sqrt{\kappa}}$
• The theorem establishes a fundamental link between curvature and topology, showing that positive Ricci curvature implies compactness
• Bonnet-Myers theorem has significant implications for the global geometry and topology of Riemannian manifolds
  • It provides a sufficient condition for a manifold to be compact, which is a crucial property in many geometric and topological arguments
• The diameter bound in the Bonnet-Myers theorem is sharp, as demonstrated by the sphere with constant sectional curvature
• The theorem can be seen as a partial converse to the Hadamard-Cartan theorem, which states that complete simply-connected manifolds with non-positive sectional curvature are diffeomorphic to Euclidean space
• Bonnet-Myers theorem has been generalized in various ways, such as the Grove-Petersen filling radius theorem and the Gromov compactness theorem

Proof Techniques and Strategies

• The original proof of the Bonnet-Myers theorem by Bonnet used Jacobi field estimates and the second variation formula for arc length
• Modern proofs often rely on the Rauch comparison theorem and the index form for geodesics
  • The index form relates the Ricci curvature along a geodesic to the growth of Jacobi fields
• A key step in the proof is to show that if the manifold is non-compact, then there exists a geodesic without conjugate points
  • This is done by constructing a sequence of geodesics minimizing length between points moving off to infinity and using the Arzelà-Ascoli theorem to extract a convergent subsequence
• The absence of conjugate points along the limit geodesic contradicts the Rauch comparison theorem, implying that the manifold must be compact
• The diameter bound is obtained by applying the Rauch comparison theorem to a minimizing geodesic between two points realizing the diameter
• Alternative proofs using Gromov's compactness theorem or the Cheeger-Gromoll splitting theorem showcase the interconnectedness of key results in Riemannian geometry

Applications in Riemannian Geometry

• The Bonnet-Myers theorem is a fundamental tool in the study of complete Riemannian manifolds with positive Ricci curvature
• It is used to prove the Cheeger-Gromoll splitting theorem, which states that a complete Riemannian manifold with non-negative Ricci curvature and a line (a complete geodesic minimizing distance between any two of its points) must split isometrically as a product of the line and a compact manifold
• The theorem plays a crucial role in the classification of compact manifolds with positive Ricci curvature
  • In low dimensions, it is known that such manifolds must be diffeomorphic to spherical space forms
• Bonnet-Myers theorem is used in the study of Einstein manifolds, which are Riemannian manifolds with constant Ricci curvature
  • In particular, it implies that compact Einstein manifolds with positive scalar curvature must have finite fundamental group
• The theorem has applications in comparison geometry, where it is used to derive various geometric and topological properties of manifolds by comparing them to spaces of constant curvature
• Bonnet-Myers theorem is also relevant in mathematical physics, particularly in the study of general relativity and the geometry of spacetime

Related Theorems and Extensions

• Synge's theorem a strengthening of the Bonnet-Myers theorem for even-dimensional manifolds with positive sectional curvature, showing that such manifolds must be simply-connected
• Grove-Petersen filling radius theorem a generalization of the Bonnet-Myers theorem, relating the filling radius (a geometric invariant measuring the size of "holes" in a manifold) to a lower bound on the Ricci curvature
• Gromov compactness theorem an extension of the Bonnet-Myers theorem to sequences of Riemannian manifolds with a uniform lower bound on Ricci curvature, showing that such sequences have a subsequence converging in the Gromov-Hausdorff sense to a compact metric space
• Cheeger-Gromoll splitting theorem a structure theorem for complete Riemannian manifolds with non-negative Ricci curvature, generalizing the Bonnet-Myers theorem
• Cheng's maximal diameter theorem a sharpening of the Bonnet-Myers diameter bound for complete Riemannian manifolds with Ricci curvature bounded below by $(n-1)$
• Brendle-Schoen classification theorem a classification of compact manifolds with 1/4-pinched sectional curvature, using the Bonnet-Myers theorem and the Ricci flow
• Perelman's proof of the Poincaré conjecture a landmark result in geometric topology, utilizing the Bonnet-Myers theorem and the Ricci flow to characterize 3-dimensional spherical space forms

Exercises and Problem-Solving Approaches

• Verify the sharpness of the Bonnet-Myers diameter bound by computing the diameter of the sphere with constant sectional curvature
• Prove that a compact Riemannian manifold with positive Ricci curvature has finite fundamental group using the Bonnet-Myers theorem and the theory of covering spaces
• Use the Bonnet-Myers theorem to show that a complete Riemannian manifold with Ricci curvature bounded below by a positive constant has a finite number of ends
• Prove the Cheeger-Gromoll splitting theorem using the Bonnet-Myers theorem and the Busemann function associated with a line in the manifold
• Apply the Bonnet-Myers theorem to show that a compact Einstein manifold with positive scalar curvature must have finite fundamental group
• Use the Gromov compactness theorem to prove the existence of a convergent subsequence for a sequence of Riemannian manifolds with a uniform lower bound on Ricci curvature and a uniform upper bound on diameter
• Prove Cheng's maximal diameter theorem using the Bonnet-Myers theorem and the Laplacian comparison theorem for the distance function from a point
• Explore the relationship between the Bonnet-Myers theorem and the Ricci flow, and how the theorem is used in the proof of the Poincaré conjecture