On tori without conjugate points
Tóm tắt
In this paper we consider Riemannian metrics without conjugate points on an n-torus. Recent work of J. Heber established that the gradient vector fields of Busemann functions on the universal cover of such a manifold induce a natural foliation (akin to the weak stable foliation for a Riemannian manifold with negative sectional curvature) on the unit tangent bundle. The main result in the paper is that the metric is flat if this foliation is Lipschitz. We also prove that this foliation is Lipschitz if and only if the metric has bounded asymptotes. This confirms a conjecture of E. Hopf in this case.
Tài liệu tham khảo
A. Avez: Variétés riemanniennes sans points focaux, C.R. Acad. Sci. Paris, Sér. A-B,270, A188-A191 (1970)
W. Ballmann. Nonpositively curved manifolds of higher rank, Ann. Math.,122, 597–609 (1985)
W. Ballmann, M. Brin, K. Burns: On surfaces with no conjugate points. J. Differ. Geom.,25, no. 2, 249–273 (1987)
W. Ballmann, M. Gromov, V. Schroeder: Manifolds of nonpositive curvature, Progress in Mathematics vol. 61, Birkhäuser, (1985)
K. Burns, G. Knieper: Rigidity of surfaces with no conjugate points, J. Differ. Geom.,34, 623–650 (1991)
K. Burns, R. Spatzier: Manifolds of nonpositive curvature and their buildings, Pub. I.H.E.S.,65, 35–59 (1987)
H. Busemann: The Geometry of Geodesics, Academic Press: New York (1955)
J. Cheeger, D. Ebin: Comparison theorems in Riemannian geometry, North-Holland, (1975)
C. Croke: Rigidity and the distance boundary points, J. Differ. Geom.,33, 445–464 (1991)
C. Croke: Simply connected manifolds with no conjugate points which are flat outside a compact set, Proc. Am. Math. Soc.,111, 297–298 (1991)
C. Croke: Volumes of balls in manifolds without conjugate points, Int. J. Math.,3 455–467 (1992)
C. Croke, A. Fathi: An inequality between energy and intersection, Bull. Lond. Math. Soc.,22, 489–494 (1990)
C. Croke, B. Kleiner: A rigidity theorem for simply connected manifolds without conjugate points (Preprint)
C. Croke, V. Schroeder: The fundamental group of compact manifolds without conjugate points, Comment. Math. Helv.61, 161–175 (1986)
J. Eschenburg: Horospheres and the stable part of the geodesic flow, Math. Z.153, 237–251 (1977)
J. Eschenburg, J. O'Sullivan: Growth of Jacobi fields and divergence of geodesics, Math. Z.,150, 221–237 (1976)
A. Freire, R. Mané: On the entropy of the geodesic flow in manifolds without conjugate points, Invent. Math.,69, 375–392 (1982)
L. Green: A theorem of E. Hopf, Mich. Math. J.,5, 31–34 (1958)
L. Green, R. Gulliver: Planes without conjugate points, J. Differ. Geom.,22, 43–47 (1985)
R. Greene, H.H. Wu, On a new gap phenomenon in Riemannian Geometry, Proc. Nat. Acad. Sci.,79, no. 2, part 1, 714–715 (1982)
D. Gromoll, J. Wolf: Some relations between the metric structure and the algebraic structure of the fundamental group in manifolds of nonpositive curvature, Bull. A.M.S.,77, 545–552 (1971)
M. Gromov: Volume and bounded cohomology, Pub. I.H.E.S., 56, 5–99 (1982)
J. Heber: On the geodesic flow of tori without conjugate points (to appear in Math Z).
G. Hedlund, M. Morse: Manifolds without conjugate points, Trans. Am. Math. Soc.,51, 362–386 (1942)
E. Hopf: Closed surfaces without conjugate points, Proc. Nat. Acad. Sci.,34, 47–51 (1948)
B. Kleiner Fundamental groups of manifolds without conjugate points
W. Klingenberg: Riemannian manifolds with geodesic flow of Anosov type, Ann. Math.99, 1–13 (1974)
A. Knuaf: Closed orbits and converse KAM theory, Nonlinearity,3, 961–973 (1990)
H.B. Lawson, S.T. Yau: Compact manifolds of nonpositive curvature, J. Differ. Geom.,7, 211–288 (1972)
R. Mané: Ergodic theory and Differentiable Dynamics, Springer: Berlin Heidelberg New York (1987)
J. Moser, On the volume elements on a manifold, Trans. Am. Math. Soc.,120 286–294 (1965)
J. O'Sullivan: Manifolds without conjugate points, Math. Ann.210, 295–311 (1974)
J. Palis, W. de Melo: Geometric Theory of Dynamical Systems, Springer: Berlin Heidelberg New York (1982)
L. Santaló, Un invariante afin para los cuerpos convexos del espacio de n dimensions, Portugal. Math.,8, 155–161 (1949)
W. Vannini: A note on manifolds without conjugate points of cohomogeneity two, Indiana Univ. Math. J.38, 137–140 (1989)