Locally symmetric spaces of non-negative curvature

2010 ◽  
pp. 328-336
Keyword(s):  
Negative Curvature ◽  
Symmetric Spaces ◽  
Locally Symmetric Spaces

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.

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