The Integral Formulae, introduced in chapter 1, is a useful for problems such as: prescribing higher mean curvatures of foliations, minimizing volume and energy defined for vector or plane fields on manifolds, and existence of foliations whose leaves enjoy given geometric properties. The Integral Formulae steams from a Reeb formula, for foliations on space forms which generalize the classical ones. For a special auxiliary functions the formulae involve the Newton transformations of the Weingarten operator.
The central topic of this book is Extrinsic Geometric Flow (EGF) on foliated manifolds, which may be a tool for prescribing extrinsic geometric properties of foliations. To develop EGF, one needs Variational Formulae, revealed in chapter 2, which expresses a change in different extrinsic geometric quantities of a fixed foliation under leaf-wise variation of the Riemannian Structure of the ambient manifold. Chapter 3 defines a general notion of EGF and studies the evolution of Riemannian metrics along the trajectories of this flow(e.g., describes the short-time existence and uniqueness theory and estimate the maximal existence time).Some special solutions (called Extrinsic Geometric Solutions) of EGF are presented and are of great interest, since they provide Riemannian Structures with very particular geometry of the leaves.
This work is aimed at those who have an interest in the differential geometry of submanifolds and foliations of Riemannian manifolds.
The themes of linearization, (re) integration, and global versus local calculus are emphasized throughout. Additional features include a treatment of the elements of multivariable calculus, formulated to adapt readily to the global context, an exploration of bundle theory, and a further (optional) development of Lie theory than is customary in textbooks at this level. New to the second edition is a detailed treatment of covering spaces and the fundamental group.
Students, teachers and professionals in mathematics and mathematical physics should find this a most stimulating and useful text.
"This is a carefully written and wide-ranging textbook suitable mainly for graduate courses, although some advanced undergraduate courses may benefit from the early chapters. The subject matter is differential topology and geometry, that is, the study of curves, surfaces and manifolds where the assumption of differentiability adds the tools of differentiable and integral calculus to those of topology. Within this area, the book is unusually comprehensive.... The style is clear and precise, and this makes the book a good reference text. There are many good exercises."—The Mathematical Gazette (Review of the Second Edition)
"This textbook, probably the best introduction to differential geometry to be published since Eisenhart's, greatly benefits from the author's knowledge of what to avoid, something that a beginner is likely to miss. The presentation is smooth, the choice of topics optimal, and the book can be profitably used for self teaching."—The Bulletin of Mathematical Books (Review of the First Edition)
"The author has very well succeeded in writing an interesting, stimulating and pleasant reading book [with] an intelligent equilibrium between rigor and informal."—Zentralblatt Math (Review of the First Edition)
This Work is based on the second international workshop entitled "Geometry and Symbolic Computations," held on May 15-18, 2013 at the University of Haifa and is dedicated to modeling (using symbolic calculations) in differential geometry and its applications in fields such as computer science, tomography and mechanics. It is intended to create a forum for students and researchers in pure and applied geometry to promote discussion of modern state-of-the-art in geometric modeling using symbolic programs such as MapleTM and Mathematica® , as well as presentation of new results.