Geometry Of Hypersurfaces
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Author |
: Thomas E. Cecil |
Publisher |
: Springer |
Total Pages |
: 601 |
Release |
: 2015-10-30 |
ISBN-10 |
: 9781493932467 |
ISBN-13 |
: 1493932462 |
Rating |
: 4/5 (67 Downloads) |
Synopsis Geometry of Hypersurfaces by : Thomas E. Cecil
This exposition provides the state-of-the art on the differential geometry of hypersurfaces in real, complex, and quaternionic space forms. Special emphasis is placed on isoparametric and Dupin hypersurfaces in real space forms as well as Hopf hypersurfaces in complex space forms. The book is accessible to a reader who has completed a one-year graduate course in differential geometry. The text, including open problems and an extensive list of references, is an excellent resource for researchers in this area. Geometry of Hypersurfaces begins with the basic theory of submanifolds in real space forms. Topics include shape operators, principal curvatures and foliations, tubes and parallel hypersurfaces, curvature spheres and focal submanifolds. The focus then turns to the theory of isoparametric hypersurfaces in spheres. Important examples and classification results are given, including the construction of isoparametric hypersurfaces based on representations of Clifford algebras. An in-depth treatment of Dupin hypersurfaces follows with results that are proved in the context of Lie sphere geometry as well as those that are obtained using standard methods of submanifold theory. Next comes a thorough treatment of the theory of real hypersurfaces in complex space forms. A central focus is a complete proof of the classification of Hopf hypersurfaces with constant principal curvatures due to Kimura and Berndt. The book concludes with the basic theory of real hypersurfaces in quaternionic space forms, including statements of the major classification results and directions for further research.
Author |
: Andreas Hochenegger |
Publisher |
: Springer Nature |
Total Pages |
: 297 |
Release |
: 2019-10-08 |
ISBN-10 |
: 9783030186388 |
ISBN-13 |
: 3030186385 |
Rating |
: 4/5 (88 Downloads) |
Synopsis Birational Geometry of Hypersurfaces by : Andreas Hochenegger
Originating from the School on Birational Geometry of Hypersurfaces, this volume focuses on the notion of (stable) rationality of projective varieties and, more specifically, hypersurfaces in projective spaces, and provides a large number of open questions, techniques and spectacular results. The aim of the school was to shed light on this vast area of research by concentrating on two main aspects: (1) Approaches focusing on (stable) rationality using deformation theory and Chow-theoretic tools like decomposition of the diagonal; (2) The connection between K3 surfaces, hyperkähler geometry and cubic fourfolds, which has both a Hodge-theoretic and a homological side. Featuring the beautiful lectures given at the school by Jean-Louis Colliot-Thélène, Daniel Huybrechts, Emanuele Macrì, and Claire Voisin, the volume also includes additional notes by János Kollár and an appendix by Andreas Hochenegger.
Author |
: John P. D'Angelo |
Publisher |
: Routledge |
Total Pages |
: 350 |
Release |
: 2019-07-16 |
ISBN-10 |
: 9781351416719 |
ISBN-13 |
: 1351416715 |
Rating |
: 4/5 (19 Downloads) |
Synopsis Several Complex Variables and the Geometry of Real Hypersurfaces by : John P. D'Angelo
Several Complex Variables and the Geometry of Real Hypersurfaces covers a wide range of information from basic facts about holomorphic functions of several complex variables through deep results such as subelliptic estimates for the ?-Neumann problem on pseudoconvex domains with a real analytic boundary. The book focuses on describing the geometry of a real hypersurface in a complex vector space by understanding its relationship with ambient complex analytic varieties. You will learn how to decide whether a real hypersurface contains complex varieties, how closely such varieties can contact the hypersurface, and why it's important. The book concludes with two sets of problems: routine problems and difficult problems (many of which are unsolved). Principal prerequisites for using this book include a thorough understanding of advanced calculus and standard knowledge of complex analysis in one variable. Several Complex Variables and the Geometry of Real Hypersurfaces will be a useful text for advanced graduate students and professionals working in complex analysis.
Author |
: An-Min Li |
Publisher |
: Walter de Gruyter GmbH & Co KG |
Total Pages |
: 528 |
Release |
: 2015-08-17 |
ISBN-10 |
: 9783110390902 |
ISBN-13 |
: 3110390906 |
Rating |
: 4/5 (02 Downloads) |
Synopsis Global Affine Differential Geometry of Hypersurfaces by : An-Min Li
This book draws a colorful and widespread picture of global affine hypersurface theory up to the most recent state. Moreover, the recent development revealed that affine differential geometry – as differential geometry in general – has an exciting intersection area with other fields of interest, like partial differential equations, global analysis, convex geometry and Riemann surfaces. The second edition of this monograph leads the reader from introductory concepts to recent research. Since the publication of the first edition in 1993 there appeared important new contributions, like the solutions of two different affine Bernstein conjectures, due to Chern and Calabi, respectively. Moreover, a large subclass of hyperbolic affine spheres were classified in recent years, namely the locally strongly convex Blaschke hypersurfaces that have parallel cubic form with respect to the Levi-Civita connection of the Blaschke metric. The authors of this book present such results and new methods of proof.
Author |
: Udo Simon |
Publisher |
: |
Total Pages |
: 354 |
Release |
: 1991 |
ISBN-10 |
: UOM:39015034773518 |
ISBN-13 |
: |
Rating |
: 4/5 (18 Downloads) |
Synopsis Introduction to the Affine Differential Geometry of Hypersurfaces by : Udo Simon
Author |
: Daniel Huybrechts |
Publisher |
: Cambridge University Press |
Total Pages |
: 462 |
Release |
: 2023-06-30 |
ISBN-10 |
: 9781009279994 |
ISBN-13 |
: 1009279998 |
Rating |
: 4/5 (94 Downloads) |
Synopsis The Geometry of Cubic Hypersurfaces by : Daniel Huybrechts
Cubic hypersurfaces are described by almost the simplest possible polynomial equations, yet their behaviour is rich enough to demonstrate many of the central challenges in algebraic geometry. With exercises and detailed references to the wider literature, this thorough text introduces cubic hypersurfaces and all the techniques needed to study them. The book starts by laying the foundations for the study of cubic hypersurfaces and of many other algebraic varieties, covering cohomology and Hodge theory of hypersurfaces, moduli spaces of those and Fano varieties of linear subspaces contained in hypersurfaces. The next three chapters examine the general machinery applied to cubic hypersurfaces of dimension two, three, and four. Finally, the author looks at cubic hypersurfaces from a categorical point of view and describes motivic features. Based on the author's lecture courses, this is an ideal text for graduate students as well as an invaluable reference for researchers in algebraic geometry.
Author |
: Hussein Mourtada |
Publisher |
: Birkhäuser |
Total Pages |
: 232 |
Release |
: 2017-05-16 |
ISBN-10 |
: 3319477781 |
ISBN-13 |
: 9783319477787 |
Rating |
: 4/5 (81 Downloads) |
Synopsis Algebraic Geometry and Number Theory by : Hussein Mourtada
This lecture notes volume presents significant contributions from the “Algebraic Geometry and Number Theory” Summer School, held at Galatasaray University, Istanbul, June 2-13, 2014. It addresses subjects ranging from Arakelov geometry and Iwasawa theory to classical projective geometry, birational geometry and equivariant cohomology. Its main aim is to introduce these contemporary research topics to graduate students who plan to specialize in the area of algebraic geometry and/or number theory. All contributions combine main concepts and techniques with motivating examples and illustrative problems for the covered subjects. Naturally, the book will also be of interest to researchers working in algebraic geometry, number theory and related fields.
Author |
: Alexandru Dimca |
Publisher |
: Springer Science & Business Media |
Total Pages |
: 277 |
Release |
: 2012-12-06 |
ISBN-10 |
: 9781461244042 |
ISBN-13 |
: 1461244048 |
Rating |
: 4/5 (42 Downloads) |
Synopsis Singularities and Topology of Hypersurfaces by : Alexandru Dimca
Author |
: Alexandru Dimca |
Publisher |
: Springer Science & Business Media |
Total Pages |
: 284 |
Release |
: 1992-04-29 |
ISBN-10 |
: 0387977090 |
ISBN-13 |
: 9780387977096 |
Rating |
: 4/5 (90 Downloads) |
Synopsis Singularities and Topology of Hypersurfaces by : Alexandru Dimca
From the very beginning, algebraic topology has developed under the influ ence of the problems posed by trying to understand the topological properties of complex algebraic varieties (e.g., the pioneering work by Poincare and Lefschetz). Especially in the work of Lefschetz [Lf2], the idea is made explicit that singularities are important in the study of the topology even in the case of smooth varieties. What is known nowadays about the topology of smooth and singular vari eties is quite impressive. The many existing results may be roughly divided into two classes as follows: (i) very general results or theories, like stratified Morse theory and (mixed) Hodge theory, see, for instance, Goresky-MacPherson [GM], Deligne [Del], and Steenbrink [S6]; and (ii) specific topics of great subtlety and beauty, like the study of the funda mental group of the complement in [p>2 of a singular plane curve initiated by Zariski or Griffiths' theory relating the rational differential forms to the Hodge filtration on the middle cohomology group of a smooth projec tive hypersurface. The aim of this book is precisely to introduce the reader to some topics in this latter class. Most of the results to be discussed, as well as the related notions, are at least two decades old, and specialists use them intensively and freely in their work. Nevertheless, it is impossible to find an adequate intro duction to this subject, which gives a good feeling for its relations with other parts of algebraic geometry and topology.
Author |
: Alexander Isaev |
Publisher |
: Springer Science & Business Media |
Total Pages |
: 231 |
Release |
: 2011-03-31 |
ISBN-10 |
: 9783642197826 |
ISBN-13 |
: 3642197825 |
Rating |
: 4/5 (26 Downloads) |
Synopsis Spherical Tube Hypersurfaces by : Alexander Isaev
We consider Levi non-degenerate tube hypersurfaces in complex linear space which are "spherical", that is, locally CR-equivalent to the real hyperquadric. Spherical hypersurfaces are characterized by the condition of the vanishing of the CR-curvature form, so such hypersurfaces are flat from the CR-geometric viewpoint. On the other hand, such hypersurfaces are of interest from the point of view of affine geometry. Thus our treatment of spherical tube hypersurfaces in this book is two-fold: CR-geometric and affine-geometric. Spherical tube hypersurfaces turn out to possess remarkable properties. For example, every such hypersurface is real-analytic and extends to a closed real-analytic spherical tube hypersurface in complex space. One of our main goals is to give an explicit affine classification of closed spherical tube hypersurfaces whenever possible. In this book we offer a comprehensive exposition of the theory of spherical tube hypersurfaces starting with the idea proposed in the pioneering work by P. Yang (1982) and ending with the new approach due to G. Fels and W. Kaup (2009).