1 What Is Curvature?

The Euclidean Plane

Surfaces in Space

Curvature in Higher Dimensions

2 Review of Tensors,Manifolds,and Vector Bundles

Tensors on a Vector Space

Manifolds

Vector Bundles

Tensor Bundles and Tensor Fields

3 Definitions and Examples of Riemannian Metrics

Riemannian Metrics

Elementary Constructions Associated with Riemannian Metrics

Generalizations of Riemannian Metrics

The Model Spaces of Riemannian Geometry

Problems

4 Connections

The Problem of Differentiating Vector Fields

Connections

Vector Fields Along Curves

Geodesics

Problems

5 Riemannian Geodesics

The Riemannian Connection

The Exponential Map

Normal Neighborhoods and Normal Coordinates

Geodesics of the Model Spaces

Problems

6 Geodesics and Distance

Lengths and Distances on Riemannian Manifolds

Geodesics and Minimizing Curves

Completeness

Problems

7 Curvature

Local Invariants

Flat Manifolds

Symmetries of the Curvature Tensor

Ricci and Scalar Curvatures

Problems

8 Riemannian Submanifolds

Riemannian Submanifolds and the Second Fundamental Form

Hypersurfaces in Euclidean Space

Geometric Interpretation of Curvature in Higher Dimensions

Problems

9 The Gauss-Bonnet Theorem

Some Plane Geometry

The Gauss-Bonnet Formula

The Gauss-Bonnet Theorem

Problems

10 Jacobi Fields

The Jacobi Equation

Computations of Jacobi Fields

Conjugate Points

The Second Variation Formula

Geodesics Do Not Minimize Past Conjugate Points

Problems

11 Curvature and Topology

Some Comparison Theorems

Manifolds of Negative Curvature

Manifolds of Positive Curvature

Manifolds of Constant Curvature

Problems

References

Index