Rigidity of some Weyl manifolds with nonpositive sectional curvature
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- by Maciej P. Wojtkowski PDF
- Proc. Amer. Math. Soc. 133 (2005), 3395-3402
Abstract:
We provide a list of all locally metric Weyl connections with nonpositive sectional curvatures on two types of manifolds, $n$-dimensional tori $\mathbb {T}^{n}$ and $\mathbb {M}^{n} =\mathbb {S}^{1}\times \mathbb {S}^{n-1}$ with the standard conformal structures. For $\mathbb {M}^{n}$ we prove that it carries no other Weyl connections with nonpositive sectional curvatures, locally metric or not. In the case of $\mathbb {T}^{n}$ we prove the same in the more narrow class of integrable connections.References
- Robert Azencott and Edward N. Wilson, Homogeneous manifolds with negative curvature. I, Trans. Amer. Math. Soc. 215 (1976), 323–362. MR 394507, DOI 10.1090/S0002-9947-1976-0394507-4
- Gerald B. Folland, Weyl manifolds, J. Differential Geometry 4 (1970), 145–153. MR 264542
- Giovanni Gallavotti and David Ruelle, SRB states and nonequilibrium statistical mechanics close to equilibrium, Comm. Math. Phys. 190 (1997), no. 2, 279–285. MR 1489572, DOI 10.1007/s002200050241
- Paul Gauduchon, La $1$-forme de torsion d’une variété hermitienne compacte, Math. Ann. 267 (1984), no. 4, 495–518 (French). MR 742896, DOI 10.1007/BF01455968
- David Ruelle, Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics, J. Statist. Phys. 95 (1999), no. 1-2, 393–468. MR 1705592, DOI 10.1023/A:1004593915069
- Maciej P. Wojtkowski, $W$-flows on Weyl manifolds and Gaussian thermostats, J. Math. Pures Appl. (9) 79 (2000), no. 10, 953–974. MR 1801870, DOI 10.1016/S0021-7824(00)00176-8
- Maciej P. Wojtkowski, Weyl manifolds and Gaussian thermostats, Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 511–523. MR 1957557
Additional Information
- Maciej P. Wojtkowski
- Affiliation: Department of Mathematics, University of Arizona, Tucson, Arizona 85721
- Email: maciejw@math.arizona.edu
- Received by editor(s): April 11, 2003
- Received by editor(s) in revised form: April 12, 2004
- Published electronically: June 20, 2005
- Additional Notes: The author is grateful to Leonid Friedlander, Feliks Przytycki, Don Wang and the referee for their comments.
- Communicated by: Jon G. Wolfson
- © Copyright 2005 by the author
- Journal: Proc. Amer. Math. Soc. 133 (2005), 3395-3402
- MSC (2000): Primary 53C99, 37Dxx
- DOI: https://doi.org/10.1090/S0002-9939-05-07809-3
- MathSciNet review: 2161165