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2018 | Book

Lorentzian Geometry: Holonomy, Spinors, and Cauchy Problems Read chapter Read first chapter

Geometric Flows and the Geometry of Space-time

Editors: Prof. Vicente Cortés, Prof. Dr. Klaus Kröncke, Jan Louis

Publisher: Springer International Publishing

Book Series : Tutorials, Schools, and Workshops in the Mathematical Sciences

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

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About this book

This book consists of two lecture notes on geometric flow equations (O. Schnürer) and Lorentzian geometry - holonomy, spinors and Cauchy Problems (H. Baum and T. Leistner) written by leading experts in these fields.

It grew out of the summer school “Geometric flows and the geometry of space-time” held in Hamburg (2016) and provides an excellent introduction for students of mathematics and theoretical physics to important themes of current research in global analysis, differential geometry and mathematical physics

Table of Contents

Frontmatter
Lorentzian Geometry: Holonomy, Spinors, and Cauchy Problems
Abstract
This review is based on lectures given by the authors during the Summer School Geometric Flows and the Geometry of Space-Time at the University of Hamburg, September 19–23, 2016. In the first part we describe the algebraic classification of connected Lorentzian holonomy groups. In particular, we specify the holonomy groups of locally indecomposable Lorentzian spin manifolds with a parallel spinor field. In the second part we explain new methods for the construction of globally hyperbolic Lorentzian manifolds with special holonomy based on the solution of certain Cauchy problems for PDEs that are imposed by the existence of a parallel lightlike vector field or a parallel lightlike spinor field with initial conditions on a spacelike hypersurface. Thereby, we derive a second order evolution equation of Cauchy-Kowalevski type that can be solved in the analytic setting as well as an appropriate first order quasilinear hyperbolic system that yields a solution in the smooth case.
Helga Baum, Thomas Leistner
Geometric Flow Equations
Abstract
In this minicourse, we study hypersurfaces that solve geometric evolution equations. More precisely, we investigate hypersurfaces that evolve with a normal velocity depending on a curvature function like the mean curvature or Gauß curvature. In three lectures, we address
  • hypersurfaces, principal curvatures and evolution equations for geometric quantities like the metric and the second fundamental form.
  • the convergence of convex hypersurfaces to round points. Here, we will also show some computer algebra calculations.
  • the evolution of graphical hypersurfaces under mean curvature flow.
Oliver C. Schnürer
Metadata
Title
Geometric Flows and the Geometry of Space-time
Editors
Prof. Vicente Cortés
Prof. Dr. Klaus Kröncke
Jan Louis
Copyright Year
2018
Electronic ISBN
978-3-030-01126-0
Print ISBN
978-3-030-01125-3
DOI
https://doi.org/10.1007/978-3-030-01126-0

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