Aspects of Differential Geometry IV (eBook)

Esteban Calviño-Louzao, Eduardo García-Río, Peter Gilkey, Jeonghyeong Park, Ramón Vázquez-Lorenzo (Autoren)

eBook Download: PDF

2019
167 Seiten
Morgan & Claypool Publishers (Verlag)
978-1-68173-564-1 (ISBN)

Infos

Book IV continues the discussion begun in the first three volumes.Although it is aimed at first-year graduate students, it is also intended to serve as a basic reference for people working in affine differential geometry. It also should be accessible to undergraduates interested in affine differential geometry. We are primarily concerned with the study of affine surfaces which are locally homogeneous. We discuss affine gradient Ricci solitons, affine Killing vector fields, and geodesic completeness. Opozda has classified the affine surface geometries which are locally homogeneous; we follow her classification. Up to isomorphism, there are two simply connected Lie groups of dimension 2. The translation group is Abelian and the ???? + ?? group is non-Abelian. The first chapter presents foundational material. The second chapter deals with Type ?? surfaces. These are the left-invariant affine geometries on . Associating to each Type ?? surface the space of solutions to the quasi-Einstein equation corresponding to the eigenvalue ?? = -1 turns out to be a very powerful technique and plays a central role in our study as it links an analytic invariant with the underlying geometry of the surface. The third chapter deals with Type ?? surfaces; these are the left-invariant affine geometries on the ???? + ?? group. These geometries form a very rich family which is only partially understood. The only remaining homogeneous geometry is that of the sphere ?? The fourth chapter presents relations between the geometry of an affine surface and the geometry of the cotangent bundle equipped with the neutral signature metric of the modified Riemannian extension.

Inhalt

• Preface


• Acknowledgments


• An Introduction to Affine Geometry


• The Geometry of Type A Models


• The Geometry of Type B Models


• Applications of Affine Surface Theory


• Bibliography


• Authors' Biographies


• Index

Produkt-Details

• Erscheint lt. Verlag: 18.4.2019
• Reihe/Serie: Synthesis Lectures on Mathematics and Statistics
• Mitarbeit: Herausgeber (Serie): Steven G. Krantz
• Verlagsort: San Rafael
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Allgemeines / Lexika
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Schlagworte: affine gradient Ricci solitons • affine Killing vector fields • geodesic completeness • locally homogeneous affine surfaces • locally symmetric affine surfaces • projectively flat • quasi-Einstein equation
• ISBN-10: 1-68173-564-4 / 1681735644
• ISBN-13: 978-1-68173-564-1 / 9781681735641