scholarly journals Higher rank rigidity for Berwald spaces

2018 ◽  
Vol 40 (7) ◽  
pp. 1991-2016
Author(s):  
WEISHENG WU
Keyword(s):  
Symmetric Space ◽  
Minkowski Space ◽  
Finite Volume ◽  
Universal Cover ◽  
Rigidity Theorem ◽  
Finsler Spaces ◽  
Locally Symmetric Space ◽  
Berwald Space ◽  
Higher Rank ◽  
Positive Flag Curvature

We generalize the higher rank rigidity theorem to a class of Finsler spaces, i.e. Berwald spaces. More precisely, we prove that a complete connected Berwald space of finite volume and bounded non-positive flag curvature with rank at least two whose universal cover is irreducible is a locally symmetric space or a locally Minkowski space.

On a conjecture of Green

1997 ◽  
Vol 17 (1) ◽  
pp. 247-252
Author(s):  
CHENGBO YUE
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Space ◽  
Sectional Curvature ◽  
Geodesic Flow ◽  
Rigidity Theorem ◽  
Locally Symmetric Space ◽  
Closed Riemannian Manifold ◽  
Rank One ◽  
The Mean ◽  
Negative Sectional Curvature

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.

scholarly journals On some compact almost Kähler locally symmetric space

1998 ◽  
Vol 21 (1) ◽  
pp. 69-72 ◽  
Author(s):  
Takashi Oguro
Keyword(s):  
Symmetric Space ◽  
Scalar Curvature ◽  
Kähler Manifold ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Locally Symmetric Space ◽  
Kahler Manifold ◽  
Almost Kähler

In the framework of studying the integrability of almost Kähler manifolds, we prove that if a compact almost Kähler locally symmetric spaceMis a weakly ,∗-Einstein vnanifold with non-negative ,∗-scalar curvature, thenMis a Kähler manifold.

scholarly journals Hyperbolic rank rigidity for manifolds of -pinched negative curvature

2018 ◽  
Vol 40 (5) ◽  
pp. 1194-1216
Author(s):  
CHRIS CONNELL ◽  
THANG NGUYEN ◽  
RALF SPATZIER
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Space ◽  
Sectional Curvature ◽  
Negative Curvature ◽  
Jacobi Field ◽  
Rigidity Result ◽  
Locally Symmetric Space ◽  
Rank One ◽  
Sectional Curvatures

A Riemannian manifold $M$ has higher hyperbolic rank if every geodesic has a perpendicular Jacobi field making sectional curvature $-1$ with the geodesic. If, in addition, the sectional curvatures of $M$ lie in the interval $[-1,-\frac{1}{4}]$ and $M$ is closed, we show that $M$ is a locally symmetric space of rank one. This partially extends work by Constantine using completely different methods. It is also a partial counterpart to Hamenstädt’s hyperbolic rank rigidity result for sectional curvatures $\leq -1$, and complements well-known results on Euclidean and spherical rank rigidity.

scholarly journals A rank rigidity result for CAT(0) spaces with one-dimensional Tits boundaries

2019 ◽  
Vol 31 (5) ◽  
pp. 1317-1330
Author(s):  
Russell Ricks
Keyword(s):  
Symmetric Space ◽  
Group Action ◽  
Spherical Building ◽  
Rigidity Result ◽  
One Dimensional ◽  
Higher Rank ◽  
Geodesically Complete ◽  
Euclidean Building ◽  
Alternate Condition

AbstractWe prove the following rank rigidity result for proper {\operatorname{CAT}(0)} spaces with one-dimensional Tits boundaries: Let Γ be a group acting properly discontinuously, cocompactly, and by isometries on such a space X. If the Tits diameter of {\partial X} equals π and Γ does not act minimally on {\partial X}, then {\partial X} is a spherical building or a spherical join. If X is also geodesically complete, then X is a Euclidean building, higher rank symmetric space, or a nontrivial product. Much of the proof, which involves finding a Tits-closed convex building-like subset of {\partial X}, does not require the Tits diameter to be π, and we give an alternate condition that guarantees rigidity when this hypothesis is removed, which is that a certain invariant of the group action be even.

A rigidity theorem for Haken manifolds

1995 ◽  
Vol 118 (1) ◽  
pp. 141-160 ◽  
Author(s):  
Teruhiko Soma
Keyword(s):  
Finite Volume ◽  
Constant Curvature ◽  
Topological Type ◽  
Riemannian Metric ◽  
Rigidity Theorem ◽  
Hyperbolic Structure ◽  
Continuous Map

A compact, orientable 3-manifold M is called hyperbolic if int M admits a complete hyperbolic structure (Riemannian metric of constant curvature − 1) of finite volume. Any hyperbolic 3-manifold M is irreducible, and each component of ∂M is an incompressible torus. Let f: M → N be a proper, continuous map between hyperbolic 3-manifolds. By Mostow's Rigidity Theorem [8], if f is π1-isomorphic then f is properly homotopic to a diffeomorphism g: M → N such that g | int M: int M → int N is isometric. In particular, the topological type of int M determines uniquely the hyperbolic structure on int M up to isometry, so the volume vol (int M) of int M is well-defined. This Rigidity Theorem is generalized by Thurston[11, theorem 6·4] so that any proper, continuous map f:M → N between hyperbolic 3-manifolds with vol(int M) = deg(f) vol(int N) is properly homotopic to a deg(f)-fold covering g:M → N such that g | int M: int M → int N is locally isometric.

scholarly journals Rigidity properties for commuting automorphisms on tori and solenoids

2021 ◽  
pp. 1-46
Author(s):  
MANFRED EINSIEDLER ◽  
ELON LINDENSTRAUSS
Keyword(s):  
Complete Classification ◽  
Rigidity Theorem ◽  
Positive Entropy ◽  
Measure Rigidity ◽  
Higher Rank ◽  
Measurable Factors

Abstract Assuming positive entropy, we prove a measure rigidity theorem for higher rank actions on tori and solenoids by commuting automorphisms. We also apply this result to obtain a complete classification of disjointness and measurable factors for these actions.

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