Geodesic flow in certain manifolds without conjugate points
Author:
Patrick Eberlein
Journal:
Trans. Amer. Math. Soc. 167 (1972), 151170
MSC:
Primary 58E10; Secondary 53C20
MathSciNet review:
0295387
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: A complete simply connected Riemannian manifold H without conjugate points satisfies the uniform Visibility axiom if the angle subtended at a point p by any geodesic of H tends uniformly to zero as the distance from p to tends uniformly to infinity. A complete manifold M is a uniform Visibility manifold if it has no conjugate points and if the simply connected covering H satisfies the uniform Visibility axiom. We derive criteria for the existence of uniform Visibility manifolds. Let M be a uniform Visibility manifold, SM the unit tangent bundle of M and the geodesic flow on SM. We prove that if every point of SM is nonwandering with respect to then is topologically transitive on SM. We also prove that if is a normal covering of M then is topologically transitive on if is topologically transitive on SM.
 [1]
D.
V. Anosov, Geodesic flows on closed Riemann manifolds with negative
curvature., Proceedings of the Steklov Institute of Mathematics, No.
90 (1967). Translated from the Russian by S. Feder, American Mathematical
Society, Providence, R.I., 1969. MR 0242194
(39 #3527)
 [2]
N.
P. Bhatia and G.
P. Szegő, Dynamical systems: Stability theory and
applications, Lecture Notes in Mathematics, No. 35, SpringerVerlag,
Berlin, 1967. MR
0219843 (36 #2917)
 [3]
P. Eberlein and B. O'Neill, Visibility manifolds (to appear).
 [4]
L.
W. Green, Geodesic instability, Proc. Amer. Math. Soc. 7 (1956), 438–448. MR 0079804
(18,148d), http://dx.doi.org/10.1090/S00029939195600798047
 [5]
L.
W. Green, Surfaces without conjugate
points, Trans. Amer. Math. Soc. 76 (1954), 529–546. MR 0063097
(16,70d), http://dx.doi.org/10.1090/S00029947195400630973
 [6]
L.
W. Green, A theorem of E. Hopf, Michigan Math. J.
5 (1958), 31–34. MR 0097833
(20 #4300)
 [7]
Marston
Morse and Gustav
A. Hedlund, Manifolds without conjugate
points, Trans. Amer. Math. Soc. 51 (1942), 362–386. MR 0006479
(3,309f), http://dx.doi.org/10.1090/S0002994719420006479X
 [8]
W.
Klingenberg, Geodätischer Fluss auf Mannigfaltigkeiten vom
hyperbolischen Typ, Invent. Math. 14 (1971),
63–82 (German). MR 0296975
(45 #6034)
 [9]
Harold
Marston Morse, A fundamental class of geodesics on
any closed surface of genus greater than one, Trans. Amer. Math. Soc. 26 (1924), no. 1, 25–60. MR
1501263, http://dx.doi.org/10.1090/S00029947192415012639
 [10]
, Instability and transitivity, J. Math. Pures Appl. (9) 14 (1935), 4971.
 [1]
 D. V. Anosov, Geodesic flows on closed Riemannian manifolds with negative curvature, Proc. Steklov Institute Math., No. 90, Amer. Math. Soc., Providence, R. I., 1969. MR 39 #3527. MR 0242194 (39:3527)
 [2]
 N. P. Bhatia and G. P. Szegö, Dynamical systems: Stability theory and applications, Lecture Notes in Math., no. 35, SpringerVerlag, Berlin, 1967, p. 122. MR 36 #2917. MR 0219843 (36:2917)
 [3]
 P. Eberlein and B. O'Neill, Visibility manifolds (to appear).
 [4]
 L. Green, Geodesic instability, Proc. Amer. Math. Soc. 7 (1956), 438448. MR 18, 148. MR 0079804 (18:148d)
 [5]
 , Surfaces without conjugate points, Trans. Amer. Math. Soc. 76 (1954), 529546. MR 0063097 (16:70d)
 [6]
 , A theorem of E. Hopf, Michigan Math. J. 5 (1958), 3134. MR 0097833 (20:4300)
 [7]
 G. Hedlund and M. Morse, Manifolds without conjugate points, Trans. Amer. Math. Soc. 51 (1942), 362386. MR 3, 309. MR 0006479 (3:309f)
 [8]
 W. Klingenberg, Geodätischer Fluss auf Mannigfaltigkeiten vom hyperbolischen Typ, Preprint, Bonn, Germany, 1970. MR 0296975 (45:6034)
 [9]
 M. Morse, A fundamental class of geodesics on any closed surface of genus greater than one, Trans. Amer. Math. Soc. 26 (1924), 2560. MR 1501263
 [10]
 , Instability and transitivity, J. Math. Pures Appl. (9) 14 (1935), 4971.
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
58E10,
53C20
Retrieve articles in all journals
with MSC:
58E10,
53C20
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197202953874
PII:
S 00029947(1972)02953874
Keywords:
Geodesic flow,
conjugate points,
nonwandering points,
topological transitivity,
uniform Visibility
Article copyright:
© Copyright 1972 American Mathematical Society
