Book Description

This book falls into two parts. The first studies those circle diffeomorphisms which commute with a diffeomorphism f whose rotation number r(f) is irrational. When f is not smoothly conjugated to a rotation, its rotation number r(f), according to Herman's theory, cannot be Diophantine. The centralizer, f, thus reflects how good are the rational and Diophantine approximations of r. The second part is dedicated to the study of biholomorphisms of a complex variable in the neighbourhood of a fixed point. In 1942 Siegel demonstrated that such a germ is analytically linearizable if its linear part is a Diophantine rotation; the arithmetic condition imposed on the rotation number was then weakened by Bruno. A geometrical approach to the problem yields an alternative proof of this result and also demonstrates the converse. If the Bruno's arithmetic condition is not satisfied the corresponding quadratic polynomial is not linearizable. ... Read more

Book Description

A central problem in differential geometry is to relate algebraic properties of the Riemann curvature tensor to the underlying geometry of the manifold. The full curvature tensor is in general quite difficult to deal with. This book presents results about the geometric consequences that follow if various natural operators defined in terms of the Riemann curvature tensor (the Jacobi operator, the skew-symmetric curvature operator, the Szabo operator, and higher order generalizations) are assumed to have constant eigenvalues or constant Jordan normal form in the appropriate domains of definition.

The book presents algebraic preliminaries and various Schur type problems; deals with the skew-symmetric curvature operator in the real and complex settings and provides the classification of algebraic curvature tensors whose skew-symmetric curvature has constant rank 2 and constant eigenvalues; discusses the Jacobi operator and a higher order generalization and gives a unified treatment of the Osserman conjecture and related questions; and establishes the results from algebraic topology that are necessary for controlling the eigenvalue structures. An extensive bibliography is provided. Results are described in the Riemannian, Lorentzian, and higher signature settings, and many families of examples are displayed. ... Read more

Book Description

The author investigates the ∠n-time decay of solutions of the Klein-Gordon equation in one space dimension with a semi-infinite potential step and derives a limiting absorption principle for the wave equation in two space dimensions with zero Dirichlet conditions imposed on a half axis. These are model cases for transient waves in media with semi-infinite geometry involving interfaces and cracktips. ... Read more

Book Description

Although research in curve shortening flow has been very active for nearly 20 years, the results of those efforts have remained scattered throughout the literature. For the first time, The Curve Shortening Problem collects and illuminates those results in a comprehensive, rigorous, and self-contained account of the fundamental results.The authors present a complete treatment of the Gage-Hamilton theorem, a clear, detailed exposition of Grayson's convexity theorem, a systematic discussion of invariant solutions, applications to the existence of simple closed geodesics on a surface, and a new, almost convexity theorem for the generalized curve shortening problem.Many questions regarding curve shortening remain outstanding. With its careful exposition and complete guide to the literature, The Curve Shortening Problem provides not only an outstanding starting point for graduate students and new investigations, but a superb reference that presents intriguing new results for those already active in the field. ... Read more

Book Description

This book is a comprehensive report of recent developments in Finsler geometry and Spray geometry. Riemannian geometry and pseudo-Riemannian geometry are treated as the special case of Finsler geometry. The geometric methods developed in this subject are useful for studying some problems arising from biology, physics, and other fields.

Audience: The book will be of interest to graduate students and mathematicians in geometry who wish to go beyond the Riemannian world. Scientists in nature sciences will find the geometric methods presented useful. ... Read more

Book Description

The Surveys in Differential Geometry are supplements to the Journal of Differential Geometry, which are published by International Press. They include significant invited papers combining original research and overviews of the most current research in specific areas of interest to the growing Journal of Differential Geometry community.The survey volumes serve as continuing references, inspirations for new research, and introductions to the variety of topics of interest to differential geometers. These supplements are published annually since 1999.

In 1993, the Journal of Differential Geometry sponsored the second conference on Differential Geometry. While the first conference was held in 1990, there is still significant interest in the proceedings (published in 1991) of Surveys in Differential Geometry 1991, and there continues to be developments in the subject. For this reason, we intend to continue sponsoring this conference and the accompanying publication once every three years. The conference was held at Harvard University, coinciding with the celebration of Raoul Bott's 70th birthday. We are very grateful to all the speakers, who attracted a large audience, and to the three hundred participants, whose participation made this conference a big success. This volume (Vol.2) of Surveys in Differential Geometry is based on the lectures given during the 2nd conference of Journal of Differential Geometry. ... Read more

Book Description

This book covers classical differential geometry with modern applications to a variety of disciplines including math and science.Surfaces, curvatures, geometry of curves, holonomy and the Gauss-Bonnet Theorem, minimal surfaces and complex variables, geodesics, least area surfaces of revolution, surfaces of Delaunay, and more.For mathematicians, scientists and other professionals wish to learn how classic differential theory applies to practical situations in math, science and industry. ... Read more

Reviews (3)

clearest undergrad differential geometry text around
This is a very well-written text on modern differential geometry for undergraduates. The content of the book is similar to O'Neill's "Elementary Differential Geometry" (e.g. covariant derivatives, shape operators), but it's easier to read. There are many undergrad texts around -- O'Neill, do Carmo, Pressley -- but this one is the most lucidly written one hands-down.

Afer going through Oprea, one might like to tackle O'Neill's "Elementary Differential Geometry" and Vols 2-4 of Spivak's "Comprehensive Introduction to D.G."

Like O'Neill, Oprea develops surface theory using the shape operator. But Oprea takes shortcuts and doesn't develop the theory in quite the same generality as O'Neill does. For example, Oprea doesn't introduce differential forms and the exterior calculus. As a consequence, Oprea restricts himself to the Serret-Frenet equations whereas O'Neill introduces Cartan's structural equations -- of which Serret-Frenet is simply a special case -- as the method of moving frames in full generality. The structural equations are then used (by O'Neill) in both curve and surface theory.

Nice introduction and applications of differential geometry
I found this book to be a fine introduction to this subject. I was particularly pleased with the practical examples outlined in the book. Even though I am not extremely proficient with Maple, I found the exercises using this software provided important illustrations of applications.

Not a text for a rigorous mathematics course
This book is not to be used as a rigorous introduction to differential geometry.There are some definitions and theorems that are casuallydescribed, and the motive behind particular definitions are vague. Thosenot interested in MAPLE might find constant instructions for MAPLEannoying. Not to be completely negative, there are some good excercizes inthe text that I especially enjoyed. ... Read more

Book Description

Book Description

In view of the significance of the array manifold in array processing and array communications, the role of differential geometry as an analytical tool cannot be overemphasized. Differential geometry is mainly confined to the investigation of the geometric properties of manifolds in three-dimensional Euclidean space R3 and in real spaces of higher dimension.

Extending the theoretical framework to complex spaces, this invaluable book presents a summary of those results of differential geometry which are of practical interest in the study of linear, planar and three-dimensional array geometries. ... Read more

Book Description

The scalar-tensor theory of gravitation moved into the limelight in recent years due to developments in string theory, M-theory and "brane world" constructions. This book introduces the subject at a level suitable for both graduate students and researchers. It explores scalar fields, placing them in context with a discussion of Brans-Dicke theory, covering the cosmological constant problem, higher dimensional space-time, branes and conformal transformations. ... Read more

Reviews (1)

Mathematics of causality.
The notion of causality enters in math (both in its elementary
forms, as well as in its brand new research trends). While the
notion comes from physics, it is equally significant in math and in its applications. In math, it is present in the most basic
notions from Fourier series, as well as in its specialized non-commutative and geometric incarnations. In physics, it is crucial for the theory of waves,-- both classical and quantum. In quantum theory, for example, it serves as a basis for the construction of the Hilbert space which carries the quantum states, and it constitutes the math behind renormalization and time reflection.In engineering, it underlies our understanding of signal analysis. ... Read more

Book Description

In this volume the authors seek to illustrate how methods of differential geometry find application in the study of the topology of differential manifolds. Prerequisites are few since the authors take pains to set out the theory of differential forms and the algebra required. The reader is introduced to De Rham cohomology, and explicit and detailed calculations are present as examples. Topics covered include Mayer-Vietoris exact sequences, relative cohomology, Pioncare duality and Lefschetz's theorem. This book will be suitable for graduate students taking courses in algebraic topology and in differential topology. Mathematicians studying relativity and mathematical physics will find this an invaluable introduction to the techniques of differential geometry. ... Read more

Book Description

Singularity theory draws on many other areas of mathematics and, in turn, contributes to many areas both within and outside mathematics, including differential and algebraic geometry, knot theory, differential equations, bifurcation theory, Hamiltonian mechanics, optics, robotics and computer vision. This volume consists of two dozen articles from some of the best known figures in singularity theory, and it presents an up-to-date survey of research in this area. ... Read more

Book Description

Tight and taut manifolds form an important and special class of surfaces within differential geometry.This book contains in-depth articles by experts in the field as well as an extensive and comprehensive bibliography. This survey will open new avenues for further research and will be an important addition to any geometer's library. ... Read more

Book Description

Book Description

For those working in singularity theory or other areas of complex geometry, this volume will open the door to the study of Frobenius manifolds. In the first part Hertling explains the theory of manifolds with a multiplication on the tangent bundle. He then presents a simplified explanation of the role of Frobenius manifolds in singularity theory along with all the necessary tools and several applications. Readers will benefit from this careful and sound study of the fundamental structures and results in this exciting branch of mathematics. ... Read more