scholarly journals The nullity of a compact minimal hypersurface in a compact symmetric space of rank one

2004 ◽  
Vol 33 (2) ◽  
pp. 429-441
Author(s):  
Tohru GOTOH
Keyword(s):  
Symmetric Space ◽  
Minimal Hypersurface ◽  
Compact Symmetric Space ◽  
Rank One

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 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 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.

SYMMETRIC SPACE TWO-MATRIX MODELS

1992 ◽  
Vol 07 (23) ◽  
pp. 5781-5796
Author(s):  
ARLEN ANDERSON
Keyword(s):  
Ising Model ◽  
Partition Function ◽  
Symmetric Space ◽  
Matrix Models ◽  
Explicit Expression ◽  
Lie Algebras ◽  
Spherical Function ◽  
Scaling Limits ◽  
Maximum Order ◽  
Rank One

The radial form of the partition function of a two-matrix model is formally given in terms of a spherical function for matrices representing any Euclidean symmetric space. An explicit expression is obtained by constructing the spherical function by the method of intertwining. The reduction of two-matrix models based on Lie algebras is an elementary application. A model based on the rank one symmetric space isomorphic to RN is less trivial and is treated in detail. This model may be interpreted as an Ising model on a random branched polymer. It has the unusual feature that the maximum order of criticality is different in the planar and double-scaling limits.

CLOSED MINIMAL HYPERSURFACES IN COMPACT SYMMETRIC SPACES

1992 ◽  
Vol 03 (05) ◽  
pp. 629-651 ◽  
Author(s):  
CLAUDIO GORODSKI
Keyword(s):  
Symmetric Spaces ◽  
Local Geometry ◽  
Minimal Hypersurfaces ◽  
Compact Symmetric Space ◽  
Orbital Geometry ◽  
Rank One ◽  
Minimal Cones ◽  
Area Minimizing ◽  
Simply Connected ◽  
Minimal Embedding

W.Y. Hsiang, W.T. Hsiang and P. Tomter conjectured that every simply-connected, compact symmetric space of dimension ≥4 must contain some minimal hypersurfaces of sphere type. With the aid of equivariant differential geometry, they showed that this is in fact the case for many symmetric spaces of rank one and two. Let M be one of the symmetric spaces: Sn(1)×Sn(1)(n≥4), SU(6)/Sp(3), E6/F4, ℍP2 (quaternionic proj. plane) or CaP2 (Cayley proj. plane). We prove the existence of infmitely many immersed, minimal hypersurfaces of sphere type in M which are invariant under a certain group G of isometries of M. Following Hsiang and the others, the equivariant method is also used here to reduce the problem to an investigation of geodesics in M/G equipped with a metric (with singularities) depending only on the orbital geometry of the transformation group (G, M). However, our constructions are based on area minimizing homogeneous cones, corresponding to a corner singularity of M/G with the local geometry of nodal type; this can be viewed as a variation of some of their constructions which depended on some unstable minimal cones of focal type. We further apply the equivariant method to construct a minimal embedding of S1×Sn−1×Sn−1 into Sn(1)×Sn(1)(n≥2) and a minimal, embedded hypersurface of sphere type in [Formula: see text], ℍPn×ℍPn (n≥2) and CaP2×CaP2.

On the complete integrability of the geodesic flow of manifolds all of whose geodesics are closed

1997 ◽  
Vol 17 (6) ◽  
pp. 1359-1370
Author(s):  
CARLOS E. DURÁN
Keyword(s):  
Finite Group ◽  
Symmetric Space ◽  
Geodesic Flow ◽  
Complete Integrability ◽  
Integrals Of Motion ◽  
Completely Integrable ◽  
Rank One ◽  
A Finite Group

We show that the geodesic flow of a metric all of whose geodesics are closed is completely integrable, with tame integrals of motion. Applications to classical examples are given; in particular, it is shown that the geodesic flow of any quotient $M/\Gamma$ of a compact, rank one symmetric space $M$ by a finite group acting freely by isometries is completely integrable by tame integrals.

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.

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