26. CJM 2007 (vol 59 pp. 1245)
 Chen, Qun; Zhou, ZhenRong

On Gap Properties and Instabilities of $p$YangMills Fields
We consider the
$p$YangMills 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 $p$YangMills
connections, and the associated curvature $\rn$ the $p$YangMills
fields. In this paper, we prove gap properties and instability theorems for $p$YangMills
fields over submanifolds in $\mathbb{R}^{n+k}$ and $\mathbb{S}^{n+k}$.
Keywords:$p$YangMills field, gap property, instability, submanifold Categories:58E15, 53C05 

27. CJM 2006 (vol 58 pp. 381)
 Jakobson, Dmitry; Nadirashvili, Nikolai; Polterovich, Iosif

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)
 Fels, M. E.; Renner, A. G.

Nonreductive Homogeneous PseudoRiemannian Manifolds of Dimension Four
A method, due to \'Elie Cartan, is used to give an algebraic
classification of the nonreductive homogeneous pseudoRiemannian
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 Ricciflat. If a
fourdimensional nonreductive homogeneous pseudoRiemannian manifold
is simply connected, then it is shown to be diffeomorphic to
$\reals^4$. All metrics for the simply connected nonreductive
Einstein spaces are given explicitly. There are no nonreductive
pseudoRiemannian homogeneous spaces of dimension two and none of
dimension three with connected isotropy subgroup.
Keywords:Homogeneous pseudoRiemannian, Einstein space Category:53C30 

29. CJM 2006 (vol 58 pp. 262)
30. CJM 2005 (vol 57 pp. 1291)
 Riveros, Carlos M. C.; Tenenblat, Keti

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)
 Karigiannis, Spiro

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)
 Finster, Felix; Kraus, Margarita

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)
 Zhang, Xi

Hermitian Yang_MillsHiggs 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 wellknown 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 YangMillsHiggs metric,, holomorphic vector bundle, KÃ¤hler manifolds Categories:58E15, 53C07 

34. CJM 2004 (vol 56 pp. 776)
 Lim, Yongdo

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
CartanHadamard 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, CartanHadamard 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)
 Ni, Yilong

The Heat Kernel and Green's Function on a Manifold with Heisenberg Group as Boundary
We study the Riemannian LaplaceBeltrami 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)
 Ni, Yilong

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)
 Graczyk, P.; Sawyer, P.

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)
 Kellerhals, Ruth

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)
 Shen, Zhongmin

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
nontrivial 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)
 Akrout, H.

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)
 Nicolaescu, Liviu I.

SeibergWitten Invariants of Lens Spaces
We show that the SeibergWitten invariants of a lens space determine
and are determined by its CassonWalker invariant and its
ReidemeisterTuraev torsion.
Keywords:lens spaces, Seifert manifolds, SeibergWitten invariants, CassonWalker invariant, Reidemeister torsion, eta invariants, DedekindRademacher sums Categories:58D27, 57Q10, 57R15, 57R19, 53C20, 53C25 

42. CJM 2000 (vol 52 pp. 757)
 Hanani, Abdellah

Le problÃ¨me de Neumann pour certaines Ã©quations du type de MongeAmpÃ¨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 MongeAmp\`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 nondecreasing function in $u$.
Keywords:connexion de LeviCivita, Ã©quations de MongeAmpÃ¨re, problÃ¨me de Neumann, estimÃ©es a priori, mÃ©thode de continuitÃ© Categories:35J60, 53C55, 58G30 

43. CJM 1998 (vol 50 pp. 1298)
 Milson, Robert

Imprimitively generated Liealgebraic Hamiltonians and separation of variables
Turbiner's conjecture posits that a Liealgebraic 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 Liealgebra acts imprimitively.
The general form of the conjecture is false. A counterexample is
given based on the trigonometric OlshanetskyPerelomov potential
corresponding to the $A_2$ root system.
Categories:35Q40, 53C30, 81R05 

44. CJM 1997 (vol 49 pp. 1162)
 Ku, HsuTung; Ku, MeiChin; Zhang, XinMin

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, GaussBonnet theorem, polygon, pseudopolygon, pseudoperimeter, hyperbolic surface, Heron's formula, analytic and geometric isoperimetric inequalities Categories:51M10, 51M25, 52A40, 53C20 

45. CJM 1997 (vol 49 pp. 1323)
 Sankaran, Parameswaran; Zvengrowski, Peter

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 Ktheory), we settle these
previously undecided families and show that none of
the manifolds in them is stably parallelizable,
apart from one 30dimensional manifold which still
remains undecided.
Categories:57R25, 55N15, 53C30 

46. CJM 1997 (vol 49 pp. 696)
47. CJM 1997 (vol 49 pp. 417)
 Boe, Brian D.; Fu, Joseph H. G.

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 KazhdanLusztig
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)
 Sawyer, P.

Estimates for the heat kernel on $\SL (n,{\bf R})/\SO (n)$
In \cite{Anker}, JeanPhilippe 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 
