Offbeat Integral Geometry on Symmetric Spaces

2013 ◽  
Author(s):  
Valery V. Volchkov ◽  
Vitaly V. Volchkov
Keyword(s):  
Integral Geometry ◽  
Symmetric Spaces

scholarly journals Integral geometry under cut loci in compact symmetric spaces

1995 ◽  
Vol 137 ◽  
pp. 33-53 ◽  
Author(s):  
Hiroyuki Tasaki
Keyword(s):  
Invariant Measure ◽  
Lie Group ◽  
Integral Geometry ◽  
Symmetric Spaces ◽  
Homogeneous Spaces ◽  
Integral Invariant ◽  
Identity Component ◽  
Integral Invariants ◽  
Compact Symmetric Spaces ◽  
Group Of Isometries

The theory of integral geometry has mainly treated identities between integral invariants of submanifolds in Riemannian homogeneous spaces like as dμg(g) where M and N are submanifolds in a Riemannian homogeneous spaces of a Lie group G and I(M ∩ gN) is an integral invariant of M ∩ gN. For example Poincaré’s formula is one of typical identities in integral geometry, which is as follows. We denote by M(R2) the identity component of the group of isometries of the plane R2 with a suitable invariant measure μM(R2).

Some questions on integral geometry on noncompact symmetric spaces of higher rank

2012 ◽  
Vol 170 (2) ◽  
pp. 195-203 ◽  
Author(s):  
E. K. Narayanan ◽  
A. Sitaram
Keyword(s):  
Integral Geometry ◽  
Symmetric Spaces ◽  
Higher Rank

Mean ergodic theorem in function symmetric spaces for infinite measure

2018 ◽  
Vol 2018 (1) ◽  
pp. 35-46
Author(s):  
Vladimir Chilin ◽  
◽  
Aleksandr Veksler ◽  
Keyword(s):  
Ergodic Theorem ◽  
Symmetric Spaces ◽  
Infinite Measure ◽  
Mean Ergodic Theorem

scholarly journals THE SMOOTHNESS OF ORBITAL MEASURES ON NONCOMPACT SYMMETRIC SPACES

2021 ◽  
pp. 1-20
Author(s):  
SANJIV KUMAR GUPTA ◽  
KATHRYN E. HARE
Keyword(s):  
Symmetric Space ◽  
Symmetric Spaces ◽  
Continuous Measure ◽  
Convolution Product ◽  
Absolutely Continuous ◽  
Regular Elements ◽  
Connected Subgroup ◽  
Decay Properties ◽  
Absolutely Continuous Measure ◽  
Special Case

Abstract Let $G/K$ be an irreducible symmetric space, where G is a noncompact, connected Lie group and K is a compact, connected subgroup. We use decay properties of the spherical functions to show that the convolution product of any $r=r(G/K)$ continuous orbital measures has its density function in $L^{2}(G)$ and hence is an absolutely continuous measure with respect to the Haar measure. The number r is approximately the rank of $G/K$ . For the special case of the orbital measures, $\nu _{a_{i}}$ , supported on the double cosets $Ka_{i}K$ , where $a_{i}$ belongs to the dense set of regular elements, we prove the sharp result that $\nu _{a_{1}}\ast \nu _{a_{2}}\in L^{2},$ except for the symmetric space of Cartan class $AI$ when the convolution of three orbital measures is needed (even though $\nu _{a_{1}}\ast \nu _{a_{2}}$ is absolutely continuous).

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