On locally symmetric spaces of non-negative curvature and certain other locally homogeneous spaces

1962 ◽  
Vol 37 (1) ◽  
pp. 266-295 ◽  
Author(s):  
Joseph A. Wolf
Keyword(s):  
Negative Curvature ◽  
Symmetric Spaces ◽  
Homogeneous Spaces ◽  
Locally Symmetric Spaces ◽  
Locally Homogeneous

A second derivative formula of the Liouville entropy at spaces of constant negative curvature

1997 ◽  
Vol 17 (5) ◽  
pp. 1131-1135 ◽  
Author(s):  
GERHARD KNIEPER
Keyword(s):  
Critical Points ◽  
Compact Manifold ◽  
Negative Curvature ◽  
Symmetric Spaces ◽  
Local Maximum ◽  
Second Derivative ◽  
Constant Negative Curvature ◽  
Locally Symmetric Spaces ◽  
Derivative Formula ◽  
Total Scalar Curvature

In this paper we study a new functional on the space of metrics with negative curvature on a compact manifold. It is a linear combination of Liouville entropy and total scalar curvature. Locally symmetric spaces are critical points of this functional. We provide an explicit formula for its second derivative at metrics of constant negative curvature. In particular, this shows that a metric of constant curvature is a local maximum.

scholarly journals Non-dense orbits on homogeneous spaces and applications to geometry and number theory

2021 ◽  
pp. 1-46
Author(s):  
JINPENG AN ◽  
LIFAN GUAN ◽  
DMITRY KLEINBOCK
Keyword(s):  
Number Theory ◽  
Hausdorff Dimension ◽  
Symmetric Spaces ◽  
Homogeneous Spaces ◽  
Discrete Subgroup ◽  
Locally Symmetric Spaces ◽  
Affine Map ◽  
Integer Points ◽  
Dense Orbits ◽  
Set Of Points

Abstract Let G be a Lie group, let $\Gamma \subset G$ be a discrete subgroup, let $X=G/\Gamma $ and let f be an affine map from X to itself. We give conditions on a submanifold Z of X that guarantee that the set of points $x\in X$ with f-trajectories avoiding Z is hyperplane absolute winning (a property which implies full Hausdorff dimension and is stable under countable intersections). A similar result is proved for one-parameter actions on X. This has applications in constructing exceptional geodesics on locally symmetric spaces and in non-density of the set of values of certain functions at integer points.

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