On a conjecture of Green
Green [5] conjectured that if $M$ is a closed Riemannian manifold of negative sectional curvature such that the mean curvatures of the horospheres through each point depend only on the point, then $V$ is a locally symmetric space of rank one. He proved this in dimension two. In this paper we prove that under Green's assumption, $M$ must be asymptotically harmonic and that the geodesic flow on $M$ is $C^{\infty}$ conjugate to that of a locally symmetric space of rank one. Combining this with the recent rigidity theorem of Besson–Courtois–Gallot [1], it follows that Green's conjecture is true for all dimensions.
1988 ◽
Vol 8
(2)
◽
pp. 215-239
◽
1991 ◽
Vol 11
(4)
◽
pp. 653-686
◽
2018 ◽
Vol 2020
(5)
◽
pp. 1346-1365
◽
2011 ◽
Vol 41
(4)
◽
pp. 447-460
◽
2008 ◽
Vol 60
(6)
◽
pp. 1201-1218
◽
2007 ◽
Vol 09
(03)
◽
pp. 401-419
◽
Keyword(s):