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This book offers a comprehensive survey to date of the theory of semiparallel submanifolds. Introduced in 1985, semiparallel submanifolds have emerged as an important area of research within differential geometry and topology.
The author begins with the necessary background on symmetric and semisymmetric Riemannian manifolds, smooth manifolds in space forms, and parallel submanifolds. Semiparallel submanifolds are introduced in Chapter 4, where characterizations of their class and several subclasses are given. In later chapters Lumiste introduces the concept of main symmetric orbit and presents all known results concerning umbilic-like main symmetric orbits. Generalizations, such as k-semiparallel submanifolds and Ric-semiparallel hypersurfaces, are also studied.
Semiparallel Submanifolds in Space Forms will appeal to both researchers and graduate students.
List of contents
Preliminaries.- Submanifolds in Space Forms.- Parallel Submanifolds.- Semiparallel Submanifolds.- Normally Flat Semiparallel Submanifolds.- Semiparallel Surfaces.- Semiparallel Three-Dimensional Submanifolds.- Decomposition Theorems.- Umbilic-Likeness of Main Symmetric Orbits.- Geometric Descriptions in General.- Isometric Semiparallel Immersions of Riemannian Manifolds of Conullity Two.- Some Generalizations.
About the author
Ülo Lumiste is considered an expert in the application of Cartan's exterior calculus. He has been noted for his extensive contributions to the area of semiparallel submanifolds.
Summary
Quite simply, this book offers the most comprehensive survey to date of the theory of semiparallel submanifolds. It begins with the necessary background material, detailing symmetric and semisymmetric Riemannian manifolds, smooth manifolds in space forms, and parallel submanifolds. The book then introduces semiparallel submanifolds and gives some characterizations for their class as well as several subclasses. The coverage moves on to discuss the concept of main symmetric orbit and presents all known results concerning umbilic-like main symmetric orbits. With more than 40 published papers under his belt on the subject, Lumiste provides readers with the most authoritative treatment.
Additional text
From the reviews:
"This is a nice book concerning submanifolds of space forms with semiparallel second fundamental form … . the present book is strongly recommended to both beginner and advance researchers in the geometry of submanifolds." (Constantin Calin, Zentralblatt MATH, Vol. 1156, 2009)
“This monograph represents a comprehensive survey and a fine contribution of the author on the topic of semiparallel submanifolds … . The book is intended for graduate students and researchers working in the area of Riemannian submanifolds. The monograph … take the reader from the fundamentals of the theory to the latest research. … The book can be effectively used as a textbook for a graduate level course or a research seminar on the topic of parallel and semiparallel submanifolds … .” (Ivko Dimitric, Mathematical Reviews, Issue 2010 h)
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From the reviews:
"This is a nice book concerning submanifolds of space forms with semiparallel second fundamental form ... . the present book is strongly recommended to both beginner and advance researchers in the geometry of submanifolds." (Constantin Calin, Zentralblatt MATH, Vol. 1156, 2009)
"This monograph represents a comprehensive survey and a fine contribution of the author on the topic of semiparallel submanifolds ... . The book is intended for graduate students and researchers working in the area of Riemannian submanifolds. The monograph ... take the reader from the fundamentals of the theory to the latest research. ... The book can be effectively used as a textbook for a graduate level course or a research seminar on the topic of parallel and semiparallel submanifolds ... ." (Ivko Dimitric, Mathematical Reviews, Issue 2010 h)
