Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A $ K$-theoretic proof of the Morse index theorem in semi-Riemannian geometry

Author: Nils Waterstraat
Journal: Proc. Amer. Math. Soc. 140 (2012), 337-349
MSC (2010): Primary 58E10; Secondary 58J20, 58J30, 34L40
Published electronically: May 5, 2011
MathSciNet review: 2833544
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We give a short proof of the Morse index theorem for geodesics in semi-Riemannian manifolds by using $ K$-theory. This makes the Morse index theorem reminiscent of the Atiyah-Singer index theorem for families of selfadjoint elliptic operators.

References [Enhancements On Off] (What's this?)

  • [APS76] M.F. Atiyah, V.K. Patodi, I.M. Singer, Spectral Asymmetry and Riemannian Geometry III, Math. Proc. Cambridge Philos. Soc., 1979, 71-99. MR 0397799 (53:1655c)
  • [Fe91] B.V. Fedosov, Index Theorems, Encyclopaedia Math. Sci. 65, Partial Differential Equations VIII, 1991, 155-251. MR 1401125
  • [Hel94] A.D. Helfer, Conjugate Points on Spacelike Geodesics or Pseudo-selfadjoint Morse-Sturm-Liouville Systems, Pacific J. Math. 164, 1994, 321-350. MR 1272654 (95j:58026)
  • [FP88] P.M. Fitzpatrick, J. Pejsachowicz, The Fundamental Group of the Space of Linear Fredholm Operators and the Global Analysis of Semilinear Equations, Contemp. Math. 72, Amer. Math. Soc., 1988, 47-87. MR 956479 (89h:47097)
  • [FPR99] P.M. Fitzpatrick, J. Pejsachowicz, L. Recht, Spectral Flow and Bifurcation of Critical Points of Strongly-Indefinite Functionals-Part I: General Theory, J. Funct. Anal. 162, 1999, 52-95. MR 1674534 (2000b:58021)
  • [La95] S. Lang, Differential and Riemannian Manifolds, Grad. Texts in Math. 160, Springer, 1995. MR 1335233 (96d:53001)
  • [Mi69] J.W. Milnor, Morse Theory, Princeton Univ. Press, 1969. MR 0163331 (29:634)
  • [MPP05] M. Musso, J. Pejsachowicz, A. Portaluri, A Morse Index Theorem for Perturbed Geodesics on Semi-Riemannian Manifolds, Topol. Methods Nonlinear Anal. 25, 2005, 69-99. MR 2133393 (2006d:58013)
  • [Pe88] J. Pejsachowicz, K-Theoretic Methods in Bifurcation Theory, Contemp. Math. 72, Amer. Math. Soc., 1988, 193-206. MR 956492 (89k:58060)
  • [PT02] P. Piccione, D.V. Tausk, The Morse Index Theorem in Semi-Riemannian Geometry, Topology 41, 2002, 1123-1159. MR 1923216 (2003g:58018)
  • [RS95] J. Robbin, D. Salamon, The Spectral Flow and the Maslov Index, Bull. Lond. Math. Soc. 27, 1995, 1-33. MR 1331677 (96d:58021)
  • [Wa07] N. Waterstraat, Der Spektralindex perturbierter semi-Riemannscher Geodaeten als Windungszahl, Diploma Thesis, University of Göttingen, 2007.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 58E10, 58J20, 58J30, 34L40

Retrieve articles in all journals with MSC (2010): 58E10, 58J20, 58J30, 34L40

Additional Information

Nils Waterstraat
Affiliation: Mathematisches Institut, Georg-August-Universität Göttingen, Bunsenstraße 3-5, D-37073 Göttingen, Germany

Received by editor(s): July 7, 2010
Received by editor(s) in revised form: October 28, 2010
Published electronically: May 5, 2011
Communicated by: Varghese Mathai
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society