There is a deep and well-known relation between probabilistic objects that are studied in stochastic analysis (typically, Brownian motion) and some analytic objects (the Laplace operator). The main purpose of this book is, roughly speaking, to explore the connection between Brownian motion and analysis in the area of differential geometry (in particular, the concept of curvature).
The basic facts about stochastic differential equations on manifolds are explained in Chapter 1, the main result being the existence and uniqueness up to explosion time for the Itô equation on a manifold. Chapter 2 studies horizontal lift and stochastic development, two concepts that are central to the Eells-Elworthy-Malliavin construction of Brownian motion on a Riemannian manifold, which is thoroughly studied in Chapter 3. Chapter 4 explores the connection between the heat kernel and Brownian motion, and considers stochastic completeness, the Feller property and recurrence and transience of the heat semigroup. Other chapters study the short-time behaviour of the heat kernel and Brownian motion, and probabilistic proofs of the Gauss-Bonnet-Chern theorem and the Atiyah-Singer index theorem. Some further applications of Brownian motion to geometric problems also appear.
The book is mainly intended for probabilists interested in geometric applications, and a basic knowledge of Euclidean stochastic analysis is assumed; differential geometry is reviewed, but the reader should have a grounding in the basic definitions. It will be useful especially for advanced graduate students and researchers interested in stochastic analysis and stochastic methods in differential geometry.