Infinitesimal conjugacies and Weil-Petersson metric

Albert Fathi; L. Flaminio

Annales de l'institut Fourier (1993)

  • Volume: 43, Issue: 1, page 279-299
  • ISSN: 0373-0956

Abstract

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We study deformations of compact Riemannian manifolds of negative curvature. We give an equation for the infinitesimal conjugacy between geodesic flows. This in turn allows us to compute derivatives of intersection of metrics. As a consequence we obtain a proof of a theorem of Wolpert.

How to cite

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Fathi, Albert, and Flaminio, L.. "Infinitesimal conjugacies and Weil-Petersson metric." Annales de l'institut Fourier 43.1 (1993): 279-299. <http://eudml.org/doc/74993>.

@article{Fathi1993,
abstract = {We study deformations of compact Riemannian manifolds of negative curvature. We give an equation for the infinitesimal conjugacy between geodesic flows. This in turn allows us to compute derivatives of intersection of metrics. As a consequence we obtain a proof of a theorem of Wolpert.},
author = {Fathi, Albert, Flaminio, L.},
journal = {Annales de l'institut Fourier},
keywords = {Teichmüller; Weil-Petersson metric; deformations; compact Riemannian manifolds of negative curvature; geodesic flows},
language = {eng},
number = {1},
pages = {279-299},
publisher = {Association des Annales de l'Institut Fourier},
title = {Infinitesimal conjugacies and Weil-Petersson metric},
url = {http://eudml.org/doc/74993},
volume = {43},
year = {1993},
}

TY - JOUR
AU - Fathi, Albert
AU - Flaminio, L.
TI - Infinitesimal conjugacies and Weil-Petersson metric
JO - Annales de l'institut Fourier
PY - 1993
PB - Association des Annales de l'Institut Fourier
VL - 43
IS - 1
SP - 279
EP - 299
AB - We study deformations of compact Riemannian manifolds of negative curvature. We give an equation for the infinitesimal conjugacy between geodesic flows. This in turn allows us to compute derivatives of intersection of metrics. As a consequence we obtain a proof of a theorem of Wolpert.
LA - eng
KW - Teichmüller; Weil-Petersson metric; deformations; compact Riemannian manifolds of negative curvature; geodesic flows
UR - http://eudml.org/doc/74993
ER -

References

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  1. [An] D. ANOSOV, Geodesic flows on closed Riemannian manifolds with negative sectional curvature, english translation, Proc. Steklov Inst. Math., 90 (1967). Zbl0176.19101MR36 #7157
  2. [Bo] F. BONAHON, Bouts de variétés hyperboliques de dimension 3, Ann. of Math., 124 (1986), 71-158. Zbl0671.57008MR88c:57013
  3. [CF] C. CROKE & A. FATHI, An inequality between energy and intersection, Bull. London Math. Soc., 22 (1990), 489-494. Zbl0719.53020MR92d:58042
  4. [FT] A. FISCHER & A. TROMBA, On a purely Riemmanian proof of the structure and dimension of the unramified moduli space of a compact Riemann surface, Math. Ann., 267 (1984), 311-345. Zbl0518.32015MR85m:58045
  5. [Gh] E. GHYS, Flots d'Anosov sur les 3-variétés fibrées en cercles, Ergodic Theory Dynamical Systems, 4 (1984), 67-80. Zbl0527.58030MR86b:58098
  6. [Gr] M. GROMOV, Three remarks on the geodesic flow, preprint. 
  7. [GK] V. GUILLEMIN & D. KAZHDAN, Some inverse spectral results for negatively curved 2-manifolds, Topology, 19 (1980), 301-312. Zbl0465.58027MR81j:58082
  8. [Kl] W. KLINGENBERG, Riemannian Geometry, Walter de Gruyter, Berlin, New York, 1982. Zbl0495.53036MR84j:53001
  9. [LM] R. de la LLAVE & R. MORIYON, Invariants for smooth conjugacy of hyperbolic dynamical systems IV, Commun. Math. Phys., 116-4 (1988), 185-192. Zbl0673.58038MR90h:58064
  10. [LMM] R. de la LLAVE, J. M. MARCO & R. MORIYON, Canonical perturbation theory of Anosov systems and regularity results for Livsic cohomology equation, Ann. of Math., 123 (1986), 537-611. Zbl0603.58016MR88h:58091
  11. [La] S. LANG, SL2 (R), Graduate Texts in Mathematics 105, Springer-Verlag, Heidelberg, New York & Tokyo, 1985. Zbl0583.22001
  12. [Mo] M. MORSE, A fundamental class of geodesics on any closed surface of genus greater than one, Trans. Amer. Math. Soc., 26 (1924), 25-60. Zbl50.0466.04JFM50.0466.04
  13. [Ta] M. TAYLOR, Noncommutative harmonic analysis, Providence, American Mathematical Society, 1986. Zbl0604.43001MR88a:22021
  14. [Tr] A. TROMBA, A classical variational approach to Teichmüller theory in “Topics in calculus of variations”, ed. M. Giaquinta, Springer Lecture Notes in Mathematics, 1365, 155-185. Zbl0694.49030MR91a:32029
  15. [Wo] S. WOLPERT, Thurston's Riemannian metric for Teichmüller space, J. Differential Geom., 23 (1986), 143-174. Zbl0592.53037MR88c:32035

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