This book, Differential Geometry:Advanced Topics inCRand Pseudohermitian Geometry(Book I-D), is the fourth in a series of four books presenting a choice of advanced topicsin Cauchy–Riemann (CR) and pseudohermitian geometry, such as Fefferman metrics,global behavior of tangentialCR equations, Rossi spheres, the CR Yamabe problem on a CR manifold-with-boundary, Jacobi fields of the Tanaka–Webster connection, the theory of CR immersionsversus Lorentzian geometry. The book also discussesbounda y values of proper holomorphic maps of balls, Beltrami equations on Rossi spheres within the Koranyi–Reimann theory of quasiconformal mappings of CR manifolds, and pseudohermitian analogs to the Gauss–Ricci–Codaz i equations in the study of CR immersions between strictly pseudoconvex CR manifolds. The other three books of the series are:

     & bsp;  Differ ntial Geometry: Manifolds,Bundles, Characteristic Classes(Book I-A)

    & bsp;    /span>Differ ntial Geometry: Riemannian Geometry and Isometric Immersions(Book I-B)

    & bsp;    /span>Differ ntial Geometry:Foundations of Cauchy-Riemann and Pseudohermitian Geometry(Book I-C)

The four books belong to an ampler book project,“Differential Geometry, Partial Differential Equations, and Mathematical Physics”, by the same authors and aim todemons rate how certain portions of differential geometry (DG) and the theory of partial differentialequati ns (PDEs) apply to general relativity and (quantum) gravity theory.

These books supply some of thead hoc DGand PDEs machinery yet do not constitute a comprehensive treatise on DG or PDEs, but ratherauthor ’ choice based on their scientific (mathematical and physical) interests. These are centered around the theory ofimmersions&mdash isometric, holomorphic,andCR&mdash and pseudohermitian geometry, as devised by Sidney Martin Webster for the study ofnondegenerate CR structures, themselves a DG manifestation of the tangential CR equations.