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Metric Connections in Projective Differential Geometry

  • Michael Eastwood
  • Vladimir Matveev
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 144)

Abstract

We search for Riemannian metrics whose Levi-Civita connection belongs to a given projective class. Following Sinjukov and Mikeš, we show that such metrics correspond precisely to suitably positive solutions of a certain projectively invariant finite-type linear system of partial differential equations. Prolonging this system, we may reformulate these equations as defining covariant constant sections of a certain vector bundle with connection. This vector bundle and its connection are derived from the Cartan connection of the underlying projective structure.

Key words

Projective differential geometry metric connection tractor 

AMS(M0S) subject classifications

Primary 53A20 Secondary 58570 

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References

  1. [1]
    E. Beltrami, Rizoluzione del problema: riportare i punti di una superficie sopra un piano in modo che le linee geodetiche vengano rappresentate da linee rette, Ann. Mat. Pura Appl. (1865), 7: 185–204.Google Scholar
  2. [2]
    T.P. Branson, A. Čap, M.G. Eastwood, AND A.R. Gover, Prolongations of geometric overdetemined systems, Int. Jour. Math. (2006), 17: 641–664.zbMATHCrossRefGoogle Scholar
  3. [3]
    A. Čap, Infinitesimal automorphisms and deformations of parabolic geometries, preprint ESI 1684 (2005), Erwin Schrödinger Institute, available at http://www.esi.ac.at.Google Scholar
  4. [4]
    E. Cartan, Sur les variétés à connexion projective, Bull. Soc. Math. France (1924), 52: 205–241.zbMATHMathSciNetGoogle Scholar
  5. [5]
    M.G. Eastwood, Notes on projective differential geometry, this volume.Google Scholar
  6. [6]
    A.R. Gover AND P. Nurowski, Obstructions to conformally Einstein metrics in n dimensions, Jour. Geom. Phys. (2006), 56: 450–484.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    J. Mikeš, Geodesic mappings of affine-connected and Riemannian spaces, Jour. Math. Sci. (1996), 78: 311–333.zbMATHCrossRefGoogle Scholar
  8. [8]
    R. Penrose AND W. Rindler, Spinors and Space-time, Vol. 1, Cambridge University Press 1984.Google Scholar
  9. [9]
    N.S. Sinjukov, Geodesic mappings of Riemannian spaces (Russian), “Nauka,” Moscow 1979.Google Scholar
  10. [10]
    T.Y. Thomas, Announcement of a projective theory of affinely connected manifolds, Proc. Nat. Acad. Sci. (1925), 11: 588–589.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Michael Eastwood
    • 1
  • Vladimir Matveev
    • 2
  1. 1.Department of MathematicsUniversity of AdelaideAustralia
  2. 2.Mathematisches Institut, Fakultät für Mathematik und InformatikFriedrich-Schiller-Universität JenaJenaGermany

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