scholarly journals Almost Positive Curvature on an Irreducible Compact Rank 2 Symmetric Space

2018 ◽  
Vol 2020 (5) ◽  
pp. 1346-1365 ◽  
Author(s):  
Jason DeVito ◽  
Ezra Nance
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Space ◽  
Sectional Curvature ◽  
Lie Group ◽  
Positive Curvature ◽  
Positive Sectional Curvature ◽  
Compact Symmetric Space ◽  
Rank 2 ◽  
Set Of Points ◽  
Positively Curved

Abstract A Riemannian manifold is said to be almost positively curved if the set of points for which all two-planes have positive sectional curvature is open and dense. We show that the Grassmannian of oriented two-planes in $\mathbb{R}^{7}$ admits a metric of almost positive curvature, giving the first example of an almost positively curved metric on an irreducible compact symmetric space of rank greater than 1. The construction and verification rely on the Lie group $\mathbf{G}_{2}$ and the octonions, so do not obviously generalize to any other Grassmannians.

scholarly journals CONVOLUTIONS OF GENERIC ORBITAL MEASURES IN COMPACT SYMMETRIC SPACES

2009 ◽  
Vol 79 (3) ◽  
pp. 513-522 ◽  
Author(s):  
SANJIV KUMAR GUPTA ◽  
KATHRYN E. HARE
Keyword(s):  
Invariant Measure ◽  
Symmetric Space ◽  
Haar Measure ◽  
Symmetric Spaces ◽  
Nonempty Interior ◽  
Absolutely Continuous ◽  
Double Cosets ◽  
Compact Symmetric Space ◽  
Compact Symmetric Spaces ◽  
Dense Set

AbstractWe prove that in any compact symmetric space, G/K, there is a dense set of a1,a2∈G such that if μj=mK*δaj*mk is the K-bi-invariant measure supported on KajK, then μ1*μ2 is absolutely continuous with respect to Haar measure on G. Moreover, the product of double cosets, Ka1Ka2K, has nonempty interior in G.

scholarly journals Regularity of spherical means and localization of spherical harmonic expansions

1986 ◽  
Vol 41 (3) ◽  
pp. 287-297 ◽  
Author(s):  
Leonardo Colzani
Keyword(s):  
Symmetric Space ◽  
Spherical Harmonic ◽  
Spherical Means ◽  
Cartan Decomposition ◽  
Compact Symmetric Space ◽  
Almost Everywhere ◽  
Rank One ◽  
Regularity Results

AbstractLet G/K be a compact symmetric space, and let G = KAK be a Cartan decomposition of G. For f in L1(G) we define the spherical means f(g, t) = ∫k∫k ∫(gktk′) dk dk′, g ∈ G, t ∈ A. We prove that if f is in Lp(G), 1 ≤ p ≤ 2, then for almost every g ∈ G the functions t → f(g, t) belong to certain Soblev spaces on A. From these regularity results for the spherical means we deduce, if G/K is a compact rank one symmetric space, a theorem on the almost everywhere localization of spherical harmonic expansions of functions in L2 (G/K).

scholarly journals Reflections and symmetries in compact symmetric spaces

1988 ◽  
Vol 38 (3) ◽  
pp. 377-386 ◽  
Author(s):  
Bang-Yen Chen ◽  
Lieven Vanhecke
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Space ◽  
Symmetric Spaces ◽  
Compact Symmetric Space ◽  
Compact Symmetric Spaces

Point symmetries and reflections are two important transformations on a Riemannian manifold. In this article we study the interactions between point symmetries and reflections in a compact symmetric space when the reflections are global isometries.

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