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1. CJM 2012 (vol 65 pp. 757)
Positively Curved Riemannian Locally Symmetric Spaces are Positively Squared Distance Curved The squared distance curvature is a kind of two-point curvature the
sign of which turned out crucial for the smoothness of optimal
transportation maps on Riemannian manifolds. Positivity properties of
that new curvature have been established recently for all the simply
connected compact rank one symmetric spaces, except the Cayley
plane. Direct proofs were given for the sphere, an indirect one
via the Hopf fibrations) for the complex and quaternionic
projective spaces. Here, we present a direct proof of a property
implying all the preceding ones, valid on every positively curved
Riemannian locally symmetric space.
Keywords:symmetric spaces, rank one, positive curvature, almost-positive $c$-curvature Categories:53C35, 53C21, 53C26, 49N60 |
2. CJM 2003 (vol 55 pp. 1000)
Some Convexity Results for the Cartan Decomposition In this paper, we consider the set $\mathcal{S} = a(e^X K e^Y)$
where $a(g)$ is the abelian part in the Cartan decomposition of
$g$. This is exactly the support of the measure intervening in the
product formula for the spherical functions on symmetric spaces of
noncompact type. We give a simple description of that support in
the case of $\SL(3,\mathbf{F})$ where $\mathbf{F} = \mathbf{R}$,
$\mathbf{C}$ or $\mathbf{H}$. In particular, we show that
$\mathcal{S}$ is convex.
We also give an application of our result to the description of
singular values of a product of two arbitrary matrices with
prescribed singular values.
Keywords:convexity theorems, Cartan decomposition, spherical functions, product formula, semisimple Lie groups, singular values Categories:43A90, 53C35, 15A18 |
3. CJM 2002 (vol 54 pp. 449)
ThÃ©orÃ¨me de Vorono\"\i\ dans les espaces symÃ©triques On d\'emontre un th\'eor\`eme de Vorono\"\i\ (caract\'erisation des
maxima locaux de l'invariant d'Hermite) pour les familles de r\'eseaux
param\'etr\'ees par les espaces sym\'etriques irr\'e\-ductibles non
exceptionnels de type non compact.
We prove a theorem of Vorono\"\i\ type (characterisation of local
maxima of the Hermite invariant) for the lattices parametrized by
irreducible nonexceptional symmetric spaces of noncompact type.
Keywords:rÃ©seaux, thÃ©orÃ¨me de Vorono\"\i, espaces symÃ©triques Categories:11H06, 53C35 |
4. CJM 1997 (vol 49 pp. 359)
Estimates for the heat kernel on $\SL (n,{\bf R})/\SO (n)$ In \cite{Anker}, Jean-Philippe Anker conjectures an upper bound for the
heat kernel of a symmetric space of noncompact type. We show in this
paper that his prediction is verified for the space of positive
definite $n\times n$ real matrices.
Categories:58G30, 53C35, 58G11 |