Differential Geometry

This book, Differential Geometry: Foundations of Cauchy-Riemann and Pseudohermitian Geometry (Book I-C), is the third in a captivating series of four books presenting a choice of topics, among fundamental and more advanced, in differential geometry (DG). It studies the basic topics of the tangential Cauchy–Riemann (CR) equations, an overdetermined system of first-order equations with smooth complex-valued coefficients induced on a real hypersurface in complex Euclidean space by the ordinary CR system. The book also gives a detailed presentation of the major tool in pseudohermitian geometry, which is a canonical linear connection (or the Tanaka–Webster connection). It also explains the geometric interpretation of Webster’s pseudohermitian sectional curvature by comparing the length of a small circle in the tangent space at a point and its projection by the exponential mapping associated with the parabolic geodesics of a given strictly pseudoconvex CR manifold. The book culminates with an explanation of the role played by J.J. Kohn’s subelliptic estimates in Kohn–Hodge–de Rham theory, devoted to the study of Kohn–Rossi cohomology. The other three books of the series are: Differential Geometry 1: Manifolds, Bundle and Characteristic Classes (Book I-A). Differential Geometry 2: Riemannian Geometry and Isometric Immersions (Book I-B). Differential Geometry 4: Advanced Topics in Cauchy–Riemann and Pseudohermitian Geometry (Book I-D). The four books belong to a larger book project (differential geometry, partial differential equations, and mathematical physics) by the same authors, aiming to demonstrate how certain portions of DG and the theory of partial differential equations apply to general relativity and (quantum) gravity theory. These books supply some of the ad hoc DG machinery yet do not constitute a comprehensive treatise on DG, but rather authors’ choice based on their scientific (mathematical and physical) interests. These are centered around the theory of immersions—isometric, holomorphic, CR—and pseudohermitian geometry, as devised by Sidney Martin Webster for the study of nondegenerate CR structures, themselves a DG manifestation of the tangential CR equations.