Differential Geometry
Riemannian Geometry and Isometric Immersions (Book I-B)
Buch in deiner Nähe kaufen
Beschreibung
This book, Differential Geometry: Riemannian Geometry and Isometric Immersions (Book I-B), is the second in a captivating series of four books presenting a choice of topics, among fundamental and more advanced in differential geometry (DG). Starting with the basics of semi-Riemannian geometry, the book aims to develop the understanding of smooth 1-parameter variations of geodesics of, and correspondingly of, Jacobi fields. A few algebraic aspects required by the treatment of the Riemann–Christoffel four-tensor and sectional curvature are successively presented. Ricci curvature and Einstein manifolds are briefly discussed. The Sasaki metric on the total space of the tangent bundle over a Riemannian manifold is built, and its main properties are investigated. An important integration technique on a Riemannian manifold, related to the geometry of geodesics, is presented for further applications. The other three books of the series are
Differential Geometry 1: Manifolds, Bundle and Characteristic Classes (Book I-A)
Differential Geometry 3: Foundations of Cauchy-Riemann and Pseudohermitian Geometry (Book I-C)
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, Cauchy–Riemann (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.
This book, Differential Geometry: Riemannian Geometry and Isometric Immersions (Book I-B), is the second in a captivating series of four books presenting a choice of topics, among fundamental and more advanced in differential geometry (DG). Starting with the basics of semi-Riemannian geometry, the book aims to develop the understanding of smooth 1-parameter variations of geodesics of, and correspondingly of, Jacobi fields. A few algebraic aspects required by the treatment of the Riemann–Christoffel four-tensor and sectional curvature are successively presented. Ricci curvature and Einstein manifolds are briefly discussed. The Sasaki metric on the total space of the tangent bundle over a Riemannian manifold is built, and its main properties are investigated. An important integration technique on a Riemannian manifold, related to the geometry of geodesics, is presented for further applications. The other three books of the series are
Differential Geometry 1: Manifolds, Bundle and Characteristic Classes (Book I-A)Differential Geometry 3: Foundations of Cauchy-Riemann and Pseudohermitian Geometry (Book I-C)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, Cauchy–Riemann (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.
Discusses the theory of submanifolds, such as the Gauss and Weingarten formula and the Gauss–Codazzi–Ricci equations Reviews the Calabi’s theorem on elliptic inequalities and Sobolev inequalities in the Euclidean setting Investigates the Sasaki metric on the total space of the tangent bundle over a Riemannian manifold
Autor*in
Elisabetta Barletta
Themen in »Differential Geometry«
Riemannian Geometry Isometric Immersions Sobolev Inequalities on Submanifolds Smooth 1-parameter Variations of Geodesics Riemann–Christofel Four-tensor Ricci Curvature Einstein Manifolds Sasaki Metric Tangent Bundle
Stimmen zu »Differential Geometry«
“The book presents an active area of modern mathematics and is addressed to a wide readership of mathematicians, physicists and students pursuing undergraduate, masters and higher degrees in mathematics and mathematical physics. ... The style is that of a mathematical textbook, with proofs given in the text or as exercises. The material is illustrated by numerous examples, some of which are taken up several times to show how the methods evolve and interact.” (Ahmed Lesfari, zbMATH 1568.53001, 2025)
()