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2017 | Book

The Riemannian and Lorentzian Splitting Theorems Read chapter Read first chapter

Topics in Modern Differential Geometry

Editors: Stefan Haesen, Leopold Verstraelen

Publisher: Atlantis Press

Book Series : Atlantis Transactions in Geometry

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

A variety of introductory articles is provided on a wide range of topics, including variational problems on curves and surfaces with anisotropic curvature. Experts in the fields of Riemannian, Lorentzian and contact geometry present state-of-the-art reviews of their topics. The contributions are written on a graduate level and contain extended bibliographies. The ten chapters are the result of various doctoral courses which were held in 2009 and 2010 at universities in Leuven, Serbia, Romania and Spain.

Table of Contents

Frontmatter
The Riemannian and Lorentzian Splitting Theorems
Abstract
In these notes we are going to briefly review some of the main ideas involved in the formulation and proof of the Riemannian and Lorentzian Splitting Theorems. We will try to emphasize the similarities and differences appeared when passing from the Riemannian to the Lorentzian case, and the way in which these difficulties are overcome by the authors.
José Luis Flores
Periodic Trajectories of Dynamical Systems Having a One-Parameter Group of Symmetries
Abstract
We study a class of dynamical systems on a compact (semi-)Riemannian manifold endowed with a non trivial 1-parameter (pre-compact) group of symmetries, and we determine the existence of a class of periodic trajectories of these systems.
Roberto Giambò, Paolo Piccione
Geometry and Materials
Abstract
We give an introduction to anisotropic surface energies motivated by the study of liquid crystal interfaces. Chandrasekhar’s proof of Wulff’s Theorem is discussed. We also recall the construction of surfaces of revolution with constant anisotropic mean curvature.
Bennett Palmer
On Deciding Whether a Submanifold Is Parabolic or Hyperbolic Using Its Mean Curvature
Abstract
We are going to see how the Hessian-Index analysis of the extrinsic distance function defined on a submanifold give us a geometric description of some of its functional theoretic properties such as its parabolicity/hyperbolicity.
Vicente Palmer
Contact Forms in Geometry and Topology
Abstract
The goal of this lecture is to give an introduction to existence problems of contact structures. So, in the first Section we define the notion of contact structure, as well as some specialized contact structures. We also study the rigidity and the local behavior of such a structure. Some basic problems concerning the geometry of contact manifolds are presented in Sect. 2. The existence of contact forms is studied in the next Section. Specially in the 3–dimensional case, some classical results and the new Geiges–Gonzalo theory of contact circles and contact spheres and the classification manifolds carrying such structures are presented. Some historical considerations pointing important steps in the development of contact geometry are finally presented.
Gheorghe Pitiş
Farkas and János Bolyai
Abstract
The lives and works of Farkas and János Bolyai are presented.
Mileva Prvanović
Spectrum Estimates and Applications to Geometry

In 1867, E. Beltrami (Ann Mat Pura Appl 1(2):329–366, 1867, [12]) introduced a second order elliptic operator on Riemannian manifolds, defined by \(\Delta ={\mathrm{{div}\,}}\circ {{\mathrm {grad}\,}}\), extending the Laplace operator on \(\mathbb {R}^{n}\), called the Laplace–Beltrami operator. The Laplace–Beltrami operator became one of the most important operators in Mathematics and Physics, playing a fundamental role in differential geometry, geometric analysis, partial differential equations, probability, potential theory, stochastic process, just to mention a few. It is in important in various differential equations that describe physical phenomena such as the diffusion equation for the heat and fluid flow, wave propagation, Laplace equation and minimal surfaces.

G. Pacelli Bessa, L. Jorge, L. Mari, J. Fábio Montenegro
Some Variational Problems on Curves and Applications
Abstract
Some variational problems are revisited showing elastic curves as a key tool to find solutions to some classical problems such as Willmore surfaces, Willmore-Chen submanifolds and 2-dimensional nonlinear sigma models. To deepen on the interplay between Geometry and Physics, some Plyushchay models have been considered.
Angel Ferrández
Special Submanifolds in Hermitian Manifolds
Abstract
The geometry of submanifolds, in particular in Hermitian manifolds, is an important topic of research in Differential Geometry.
Ion Mihai
An Introduction to Certain Topics on Lorentzian Geometry
Abstract
These notes cover the content of a mini-course of three lectures I gave in March 17–20, 2008, to young researchers within the International Research School of the Simon Stevin Institute for Geometry at Katholieke Universiteit Leuven (Belgium). My main aim was providing to the students with an introduction to several research topics on Lorentzian Geometry, including background and a panoramic view of the developments throughout the time of some interesting problems. I would like to give my sincere thanks to the organizers Stefan Haesen and Johan Gielis, Simon Stevin Institute for Geometry, Netherlands, and Leopold Verstraelen, Katholieke Universiteit Leuven, Belgium, for giving me the opportunity to talk to a number of PhD students from several countries, and I hope that my lectures encourage them to face new challenges in the beautiful research area of Lorentzian Geometry.
Alfonso Romero
Metadata
Title
Topics in Modern Differential Geometry
Editors
Stefan Haesen
Leopold Verstraelen
Copyright Year
2017
Publisher
Atlantis Press
Electronic ISBN
978-94-6239-240-3
Print ISBN
978-94-6239-239-7
DOI
https://doi.org/10.2991/978-94-6239-240-3

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