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## Über dieses Buch

​This textbook is designed for a one or two semester graduate course on Riemannian geometry for students who are familiar with topological and differentiable manifolds. The second edition has been adapted, expanded, and aptly retitled from Lee’s earlier book, Riemannian Manifolds: An Introduction to Curvature. Numerous exercises and problem sets provide the student with opportunities to practice and develop skills; appendices contain a brief review of essential background material.
While demonstrating the uses of most of the main technical tools needed for a careful study of Riemannian manifolds, this text focuses on ensuring that the student develops an intimate acquaintance with the geometric meaning of curvature. The reasonably broad coverage begins with a treatment of indispensable tools for working with Riemannian metrics such as connections and geodesics. Several topics have been added, including an expanded treatment of pseudo-Riemannian metrics, a more detailed treatment of homogeneous spaces and invariant metrics, a completely revamped treatment of comparison theory based on Riccati equations, and a handful of new local-to-global theorems, to name just a few highlights.
Reviews of the first edition:
Arguments and proofs are written down precisely and clearly. The expertise of the author is reflected in many valuable comments and remarks on the recent developments of the subjects. Serious readers would have the challenges of solving the exercises and problems. The book is probably one of the most easily accessible introductions to Riemannian geometry. (M.C. Leung, MathReview)
The book’s aim is to develop tools and intuition for studying the central unifying theme in Riemannian geometry, which is the notion of curvature and its relation with topology. The main ideas of the subject, motivated as in the original papers, are introduced here in an intuitive and accessible way…The book is an excellent introduction designed for a one-semester graduate course, containing exercises and problems which encourage students to practice working with the new notions and develop skills for later use. By citing suitable references for detailed study, the reader is stimulated to inquire into further research. (C.-L. Bejan, zBMATH)

## Inhaltsverzeichnis

### Chapter 1. What Is Curvature?

The central unifying theme in current Riemannian geometry research is the notion of curvature and its relation to topology. To put the subject in perspective, this chapter addresses some very basic questions: What is curvature? What are some important theorems about it? We explore these and related questions in an informal way, without proofs.
John M. Lee

### Chapter 2. Riemannian Metrics

In this chapter we officially define Riemannian metrics, and discuss some of the basic computational techniques associated with them and some standard methods for constructing them. At the end of the chapter, we discuss some important generalizations of Riemannian metrics—most importantly, pseudo-Riemannian metrics.
John M. Lee

### Chapter 3. Model Riemannian Manifolds

Before we delve into the general theory of Riemannian manifolds, we pause to give it some substance by introducing a variety of “model Riemannian manifolds” that should help to motivate the general theory. These manifolds are distinguished by having a high degree of symmetry. We begin by describing the most symmetric model spaces of all—Euclidean spaces, spheres, and hyperbolic spaces. Then we explore some more general classes of Riemannian manifolds with symmetry.
John M. Lee

### Chapter 4. Connections

Before defining a notion of curvature that makes sense on arbitrary Riemannian manifolds, we need to study geodesics, the generalizations to Riemannian manifolds of straight lines in Euclidean space. In this chapter, we introduce a new geometric construction called a connection, which is an essential tool for defining geodesics.
John M. Lee

### Chapter 5. The Levi-Civita Connection

On each Riemannian or pseudo-Riemannian manifold, there is a unique connection determined by the metric, called the Levi-Civita connection. After defining it, we investigate the exponential map, which conveniently encodes the collective behavior of geodesics and allows us to study the way they change as the initial point and initial velocity vary.
John M. Lee

### Chapter 6. Geodesics and Distance

In this chapter, we explore the relationships among geodesics, lengths, and distances on a Riemannian manifold. One of the main goals is to show that all length-minimizing curves are geodesics, and all geodesics are locally length minimizing. Later, we prove the Hopf–Rinow theorem, which states that a connected Riemannian manifold is geodesically complete if and only if it is complete as a metric space. At the end of the chapter, we study distance functions (which express the distance to a point or other subset) and show how they can be used to construct coordinates that put a metric in a particularly simple form.
John M. Lee

### Chapter 7. Curvature

In this chapter, we begin our study of the local invariants of Riemannian metrics. Starting with the question of whether all Riemannian metrics are locally isometric, we are led to a definition of the Riemannian curvature tensor as a measure of the failure of second covariant derivatives to commute. Then we prove the main result of this chapter: a Riemannian manifold has zero curvature if and only if it is flat, or locally isometric to Euclidean space. At the end of the chapter, we explore how the curvature can be used to detect conformal flatness.
John M. Lee

### Chapter 8. Riemannian Submanifolds

This chapter has a dual purpose: first to apply the theory of curvature to Riemannian submanifolds, and then to use these concepts to derive a precise quantitative interpretation of the curvature tensor. We first define a vector-valued bilinear form called the second fundamental form, which measures the way a submanifold curves within the ambient manifold. This leads to a quantitative geometric interpretation of the curvature tensor, as an object that encodes the sectional curvatures, which are Gaussian curvatures of 2-dimensional submanifolds swept out by geodesics tangent to 2-planes in a tangent space.
John M. Lee

### Chapter 9. The Gauss–Bonnet Theorem

In this chapter, we prove our first major local-to-global theorem in Riemannian geometry: the Gauss–Bonnet theorem. The grandfather of all such theorems in Riemannian geometry, it asserts the equality of two very differently defined quantities on a compact Riemannian 2-manifold: the integral of the Gaussian curvature, which is determined by the local geometry, and $$2\pi$$ times the Euler characteristic, which is a global topological invariant.
John M. Lee

### Chapter 10. Jacobi Fields

To generalize to higher dimensions some of the geometric and topological consequences of the Gauss–Bonnet theorem, we need to develop a new approach: instead of using Stokes’s theorem and differential forms to relate the curvature to global topology as in the proof of the Gauss–Bonnet theorem, we analyze the way that curvature affects the behavior of nearby geodesics. In this chapter, we study Jacobi fields, which encode the first-order behavior of families of geodesics. We then introduce conjugate points, which are pairs of points along a geodesic where some nontrivial Jacobi field vanishes, and show that no geodesic is minimizing past its first conjugate point.
John M. Lee

### Chapter 11. Comparison Theory

The purpose of this chapter is to show how upper or lower bounds on curvature can be used to derive bounds on other geometric quantities such as lengths of tangent vectors, distances, and volumes. In the first section of the chapter, we show how the growth of Jacobi fields in a normal neighborhood is controlled by the Hessian of the radial distance function, which satisfies a first-order differential equation called a Riccati equation. We then state and prove a fundamental comparison theorem for Riccati equations. Then we derive some of the most important geometric comparison theorems that follow from the Riccati comparison theorem.
John M. Lee

### Chapter 12. Curvature and Topology

In this final chapter, we bring together most of the tools we have developed so far to prove some significant local-to-global theorems relating curvature and topology of Riemannian manifolds. The main results are (1) the Killing–Hopf theorem, which characterizes complete, simply connected manifolds with constant sectional curvature; (2) the Cartan–Hadamard theorem, which topologically characterizes complete, simply connected manifolds with nonpositive sectional curvature; and (3) Myers’s theorem, which says that a complete manifold with Ricci curvature bounded below by a positive constant must be compact and have a finite fundamental group.
John M. Lee

### Backmatter

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