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Recent Developments in Pseudo-Riemannian Geometry
ESI Lectures in Mathematics and Physics

Recent Developments in Pseudo-Riemannian Geometry

Dmitri V. Alekseevsky (School of Mathematics, The University of Edinburgh, UK)
Helga Baum (Humboldt-Universität, Berlin, Germany)

ISBN print 978-3-03719-051-7, ISBN online 978-3-03719-551-2
DOI 10.4171/051
June 2008, 549 pages, softcover, 17 x 24 cm.
58.00 Euro

This book provides an introduction to and survey of recent developments in pseudo-Riemannian geometry, including applications in mathematical physics, by leading experts in the field. Topics covered are:

  • Classification of pseudo-Riemannian symmetric spaces
  • Holonomy groups of Lorentzian and pseudo-Riemannian manifolds
  • Hypersymplectic manifolds
  • Anti-self-dual conformal structures in neutral signature and integrable systems
  • Neutral Kähler surfaces and geometric optics
  • Geometry and dynamics of the Einstein universe
  • Essential conformal structures and conformal transformations in pseudo-Riemannian geometry
  • The causal hierarchy of spacetimes
  • Geodesics in pseudo-Riemannian manifolds
  • Lorentzian symmetric spaces in supergravity
  • Generalized geometries in supergravity
  • Einstein metrics with Killing leaves
The book is addressed to advanced students as well as to researchers in differential geometry, global analysis, general relativity and string theory. It shows essential differences between the geometry on manifolds with positive definite metrics and on those with indefinite metrics, and highlights the interesting new geometric phenomena, which naturally arise in the indefinite metric case. The reader finds a description of the present state of art in the field as well as open problems, which can stimulate further research.

Keywords: Pseudo-Riemannian manifold, Lorentzian manifold, spacetime, neutral signature, holonomy group, pseudo-Riemannian symmetric space, Cahen–Wallach space, hypersymplectic manifold, anti-self-dual conformal structure, integrable system, neutral Kähler surfaces, Einstein metric, Einstein universe, essential conformal structure, conformal transformation, causal hierarchy, geodesic, supergravity, generalized geometry, generalized G-structure, Killing spinor

Further Information

Review in Zbl MATH 1139.53002

MR 2435292 (2009f:53001)