Expand all Collapse all | Results 26 - 48 of 48 |
26. CJM 2007 (vol 59 pp. 1245)
On Gap Properties and Instabilities of $p$-Yang--Mills Fields We consider the
$p$-Yang--Mills functional
$(p\geq 2)$
defined as
$\YM_p(\nabla):=\frac 1 p \int_M \|\rn\|^p$.
We call critical points of $\YM_p(\cdot)$ the $p$-Yang--Mills
connections, and the associated curvature $\rn$ the $p$-Yang--Mills
fields. In this paper, we prove gap properties and instability theorems for $p$-Yang--Mills
fields over submanifolds in $\mathbb{R}^{n+k}$ and $\mathbb{S}^{n+k}$.
Keywords:$p$-Yang--Mills field, gap property, instability, submanifold Categories:58E15, 53C05 |
27. CJM 2006 (vol 58 pp. 381)
Extremal Metric for the First Eigenvalue on a Klein Bottle The first eigenvalue of the Laplacian on a surface can be viewed
as a functional on the space of Riemannian metrics of a given
area. Critical points of this functional are called extremal
metrics. The only known extremal metrics are a round sphere, a
standard projective plane, a Clifford torus and an equilateral
torus. We construct an extremal metric on a Klein bottle. It is a
metric of revolution, admitting a minimal isometric embedding into
a sphere ${\mathbb S}^4$ by the first eigenfunctions. Also, this
Klein bottle is a bipolar surface for Lawson's
$\tau_{3,1}$-torus. We conjecture that an extremal metric for the
first eigenvalue on a Klein bottle is unique, and hence it
provides a sharp upper bound for $\lambda_1$ on a Klein bottle of
a given area. We present numerical evidence and prove the first
results towards this conjecture.
Keywords:Laplacian, eigenvalue, Klein bottle Categories:58J50, 53C42 |
28. 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 |
29. CJM 2006 (vol 58 pp. 262)
Connections on a Parabolic Principal Bundle Over a Curve The aim here is to define connections on a parabolic
principal bundle. Some applications are given.
Keywords:parabolic bundle, holomorphic connection, unitary connection Categories:53C07, 32L05, 14F05 |
30. CJM 2005 (vol 57 pp. 1291)
Dupin Hypersurfaces in $\mathbb R^5$ We study Dupin
hypersurfaces in $\mathbb R^5$ parametrized by lines of curvature, with
four distinct principal curvatures. We characterize locally a generic
family of such hypersurfaces in terms of the principal curvatures and
four vector valued functions of one variable. We show that these vector
valued functions are invariant by inversions and homotheties.
Categories:53B25, 53C42, 35N10, 37K10 |
31. CJM 2005 (vol 57 pp. 1012)
Deformations of $G_2$ and $\Spin(7)$ Structures We consider some deformations of $G_2$-structures on $7$-manifolds. We
discover a canonical way to deform a $G_2$-structure by a vector field in
which the associated metric gets ``twisted'' in some way by the
vector cross product. We present a system of partial differential
equations for an unknown vector field $w$ whose solution would
yield a manifold with holonomy $G_2$. Similarly we consider analogous
constructions for $\Spin(7)$-structures on $8$-manifolds. Some of
the results carry over directly, while others do not because of the
increased complexity of the $\Spin(7)$ case.
Keywords:$G_2 \Spin(7)$, holonomy, metrics, cross product Categories:53C26, 53C29 |
32. CJM 2005 (vol 57 pp. 708)
Curvature Estimates in Asymptotically Flat Lorentzian Manifolds We consider an asymptotically flat Lorentzian manifold of
dimension $(1,3)$. An inequality is derived which bounds the
Riemannian curvature tensor in terms of the ADM energy in the
general case with second fundamental form. The inequality
quantifies in which sense the Lorentzian manifold becomes flat in
the limit when the ADM energy tends to zero.
Categories:53C21, 53C27, 83C57 |
33. CJM 2005 (vol 57 pp. 871)
Hermitian Yang-_Mills--Higgs Metrics on\\Complete KÃ¤hler Manifolds In this paper, first, we will investigate the
Dirichlet problem for one type of vortex equation, which
generalizes the well-known Hermitian Einstein equation. Secondly,
we will give existence results for solutions of these vortex
equations over various complete noncompact K\"ahler manifolds.
Keywords:vortex equation, Hermitian Yang--Mills--Higgs metric,, holomorphic vector bundle, KÃ¤hler manifolds Categories:58E15, 53C07 |
34. 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 |
35. CJM 2004 (vol 56 pp. 590)
The Heat Kernel and Green's Function on a Manifold with Heisenberg Group as Boundary We study the Riemannian Laplace-Beltrami operator $L$ on a Riemannian
manifold with Heisenberg group $H_1$ as boundary. We calculate the heat
kernel and Green's function for $L$, and give global and small time
estimates of the heat kernel. A class of hypersurfaces in this
manifold can be regarded as approximations of $H_1$. We also restrict
$L$ to each hypersurface and calculate the corresponding heat kernel
and Green's function. We will see that the heat kernel and Green's
function converge to the heat kernel and Green's function on the
boundary.
Categories:35H20, 58J99, 53C17 |
36. CJM 2004 (vol 56 pp. 566)
Geodesics in a Manifold with Heisenberg Group as Boundary The Heisenberg group is considered as the boundary of a manifold. A class
of hypersurfaces in this manifold can be regarded as copies of the Heisenberg
group. The properties of geodesics in the interior and on the hypersurfaces
are worked out in detail. These properties are strongly related to those of
the Heisenberg group.
Keywords:Heisenberg group, Hamiltonian mechanics, geodesic Categories:53C22, 53C17 |
37. 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 |
38. CJM 2003 (vol 55 pp. 1080)
Quaternions and Some Global Properties of Hyperbolic $5$-Manifolds We provide an explicit thick and thin decomposition for oriented
hyperbolic manifolds $M$ of dimension $5$. The result implies improved
universal lower bounds for the volume $\rmvol_5(M)$ and, for $M$
compact, new estimates relating the injectivity radius and the diameter
of $M$ with $\rmvol_5(M)$. The quantification of the thin part is
based upon the identification of the isometry group of the universal
space by the matrix group $\PS_\Delta {\rm L} (2,\mathbb{H})$ of
quaternionic $2\times 2$-matrices with Dieudonn\'e determinant
$\Delta$ equal to $1$ and isolation properties of $\PS_\Delta {\rm
L} (2,\mathbb{H})$.
Categories:53C22, 53C25, 57N16, 57S30, 51N30, 20G20, 22E40 |
39. CJM 2003 (vol 55 pp. 112)
Finsler Metrics with ${\bf K}=0$ and ${\bf S}=0$ In the paper, we study the shortest time problem on a Riemannian space
with an external force. We show that such problem can be converted
to a shortest path problem on a Randers space. By choosing an
appropriate external force on the Euclidean space, we obtain a
non-trivial Randers metric of zero flag curvature. We also show that
any positively complete Randers metric with zero flag curvature must
be locally Minkowskian.
Categories:53C60, 53B40 |
40. 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 |
41. CJM 2001 (vol 53 pp. 780)
Seiberg-Witten Invariants of Lens Spaces We show that the Seiberg-Witten invariants of a lens space determine
and are determined by its Casson-Walker invariant and its
Reidemeister-Turaev torsion.
Keywords:lens spaces, Seifert manifolds, Seiberg-Witten invariants, Casson-Walker invariant, Reidemeister torsion, eta invariants, Dedekind-Rademacher sums Categories:58D27, 57Q10, 57R15, 57R19, 53C20, 53C25 |
42. CJM 2000 (vol 52 pp. 757)
Le problÃ¨me de Neumann pour certaines Ã©quations du type de Monge-AmpÃ¨re sur une variÃ©tÃ© riemannienne |
Le problÃ¨me de Neumann pour certaines Ã©quations du type de Monge-AmpÃ¨re sur une variÃ©tÃ© riemannienne Let $(M_n,g)$ be a strictly convex riemannian manifold with
$C^{\infty}$ boundary. We prove the existence\break
of classical solution for the nonlinear elliptic partial
differential equation of Monge-Amp\`ere:\break
$\det (-u\delta^i_j + \nabla^i_ju) = F(x,\nabla u;u)$ in $M$ with a
Neumann condition on the boundary of the form $\frac{\partial
u}{\partial \nu} = \varphi (x,u)$, where $F \in C^{\infty} (TM
\times \bbR)$ is an everywhere strictly positive function
satisfying some assumptions, $\nu$ stands for the unit normal
vector field and $\varphi \in C^{\infty} (\partial M \times \bbR)$
is a non-decreasing function in $u$.
Keywords:connexion de Levi-Civita, Ã©quations de Monge-AmpÃ¨re, problÃ¨me de Neumann, estimÃ©es a priori, mÃ©thode de continuitÃ© Categories:35J60, 53C55, 58G30 |
43. 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 |
44. CJM 1997 (vol 49 pp. 1162)
Isoperimetric inequalities on surfaces of constant curvature In this paper we introduce the concepts of hyperbolic and elliptic
areas and prove uncountably many new geometric isoperimetric
inequalities on the surfaces of constant curvature.
Keywords:Gaussian curvature, Gauss-Bonnet theorem, polygon, pseudo-polygon, pseudo-perimeter, hyperbolic surface, Heron's formula, analytic and geometric isoperimetric inequalities Categories:51M10, 51M25, 52A40, 53C20 |
45. 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 |
46. CJM 1997 (vol 49 pp. 696)
Geodesic flow on ideal polyhedra In this work we study the geodesic flow on $n$-dimensional ideal polyhedra
and establish classical (for manifolds of negative curvature) results
concerning the distribution of closed orbits of the flow.
Categories:57M20, 53C23 |
47. CJM 1997 (vol 49 pp. 417)
Characteristic cycles in Hermitian symmetric spaces
We give explicit combinatorial expresssions for the characteristic
cycles associated to certain canonical sheaves on Schubert varieties
$X$ in the classical Hermitian symmetric spaces: namely the
intersection homology sheaves $IH_X$ and the constant sheaves $\Bbb
C_X$. The three main cases of interest are the Hermitian symmetric
spaces for groups of type $A_n$ (the standard Grassmannian), $C_n$
(the Lagrangian Grassmannian) and $D_n$. In particular we find that
$CC(IH_X)$ is irreducible for all Schubert varieties $X$ if and only
if the associated Dynkin diagram is simply laced. The result for
Schubert varieties in the standard Grassmannian had been established
earlier by Bressler, Finkelberg and Lunts, while the computations in
the $C_n$ and $D_n$ cases are new.
Our approach is to compute $CC(\Bbb C_X)$ by a direct geometric
method, then to use the combinatorics of the Kazhdan-Lusztig
polynomials (simplified for Hermitian symmetric spaces) to compute
$CC(IH_X)$. The geometric method is based on the fundamental formula
$$CC(\Bbb C_X) = \lim_{r\downarrow 0} CC(\Bbb C_{X_r}),$$ where the
$X_r \downarrow X$ constitute a family of tubes around the variety
$X$. This formula leads at once to an expression for the coefficients
of $CC(\Bbb C_X)$ as the degrees of certain singular maps between
spheres.
Categories:14M15, 22E47, 53C65 |
48. 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 |