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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 2012 (vol 65 pp. 66)
On Flag Curvature of Homogeneous Randers Spaces In this paper we give an explicit formula for the flag curvature of
homogeneous Randers spaces of Douglas type and apply this formula to
obtain some interesting results. We first deduce an explicit formula
for the flag curvature of an arbitrary left invariant Randers metric
on a two-step nilpotent Lie group. Then we obtain a classification of
negatively curved homogeneous Randers spaces of Douglas type. This
results, in particular, in many examples of homogeneous non-Riemannian
Finsler spaces with negative flag curvature. Finally, we prove a
rigidity result that a homogeneous Randers space of Berwald type whose
flag curvature is everywhere nonzero must be Riemannian.
Keywords:homogeneous Randers manifolds, flag curvature, Douglas spaces, two-step nilpotent Lie groups Categories:22E46, 53C30 |
3. CJM 2009 (vol 61 pp. 1201)
Invariant Einstein Metrics on Some Homogeneous Spaces of Classical Lie Groups A Riemannian manifold $(M,\rho)$ is called Einstein if the metric
$\rho$ satisfies the condition \linebreak$\Ric (\rho)=c\cdot \rho$ for some
constant $c$. This paper is devoted to the investigation of
$G$-invariant Einstein metrics, with additional symmetries,
on some homogeneous spaces $G/H$ of classical groups.
As a consequence, we obtain new invariant Einstein metrics on some
Stiefel manifolds $\SO(n)/\SO(l)$.
Furthermore, we show that for any positive integer $p$ there exists a
Stiefel manifold $\SO(n)/\SO(l)$
that admits at least $p$
$\SO(n)$-invariant Einstein metrics.
Keywords:Riemannian manifolds, homogeneous spaces, Einstein metrics, Stiefel manifolds Categories:53C25, 53C30 |
4. CJM 2006 (vol 58 pp. 282)
Non-reductive Homogeneous Pseudo-Riemannian Manifolds of Dimension Four A method, due to \'Elie Cartan, is used to give an algebraic
classification of the non-reductive homogeneous pseudo-Riemannian
manifolds of dimension four. Only one case with Lorentz signature can
be Einstein without having constant curvature, and two cases with
$(2,2)$ signature are Einstein of which one is Ricci-flat. If a
four-dimensional non-reductive homogeneous pseudo-Riemannian manifold
is simply connected, then it is shown to be diffeomorphic to
$\reals^4$. All metrics for the simply connected non-reductive
Einstein spaces are given explicitly. There are no non-reductive
pseudo-Riemannian homogeneous spaces of dimension two and none of
dimension three with connected isotropy subgroup.
Keywords:Homogeneous pseudo-Riemannian, Einstein space Category:53C30 |
5. CJM 2004 (vol 56 pp. 776)
Best Approximation in Riemannian Geodesic Submanifolds of Positive Definite Matrices We explicitly describe
the best approximation in
geodesic submanifolds of positive definite matrices
obtained from involutive
congruence transformations on the
Cartan-Hadamard manifold ${\mathrm{Sym}}(n,{\Bbb R})^{++}$ of
positive definite matrices.
An explicit calculation for the minimal distance
function from the geodesic submanifold
${\mathrm{Sym}}(p,{\mathbb R})^{++}\times
{\mathrm{Sym}}(q,{\mathbb R})^{++}$ block diagonally embedded in
${\mathrm{Sym}}(n,{\mathbb R})^{++}$ is
given in terms of metric and
spectral geometric means, Cayley transform, and Schur
complements of positive definite matrices when $p\leq 2$ or $q\leq 2.$
Keywords:Matrix approximation, positive, definite matrix, geodesic submanifold, Cartan-Hadamard manifold,, best approximation, minimal distance function, global tubular, neighborhood theorem, Schur complement, metric and spectral, geometric mean, Cayley transform Categories:15A48, 49R50, 15A18, 53C3 |
6. 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 |
7. 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 |
8. CJM 1998 (vol 50 pp. 1298)
Imprimitively generated Lie-algebraic Hamiltonians and separation of variables Turbiner's conjecture posits that a Lie-algebraic Hamiltonian
operator whose domain is a subset of the Euclidean plane admits a
separation of variables. A proof of this conjecture is given in
those cases where the generating Lie-algebra acts imprimitively.
The general form of the conjecture is false. A counter-example is
given based on the trigonometric Olshanetsky-Perelomov potential
corresponding to the $A_2$ root system.
Categories:35Q40, 53C30, 81R05 |
9. CJM 1997 (vol 49 pp. 1323)
Stable parallelizability of partially oriented flag manifolds II In the first paper with the same title the authors
were able to determine all partially oriented flag
manifolds that are stably parallelizable or
parallelizable, apart from four infinite families
that were undecided. Here, using more delicate
techniques (mainly K-theory), we settle these
previously undecided families and show that none of
the manifolds in them is stably parallelizable,
apart from one 30-dimensional manifold which still
remains undecided.
Categories:57R25, 55N15, 53C30 |
10. 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 |