Expand all Collapse all | Results 26 - 50 of 77 |
26. CJM 2010 (vol 62 pp. 975)
Revisiting Tietze-Nakajima: Local and Global Convexity for Maps
A theorem of Tietze and Nakajima, from 1928, asserts that
if a subset $X$ of $\mathbb{R}^n$ is closed, connected, and locally convex,
then it is convex.
We give an analogous ``local to global convexity" theorem
when the inclusion map of $X$ to $\mathbb{R}^n$ is replaced by a map
from a topological space $X$ to $\mathbb{R}^n$ that satisfies
certain local properties.
Our motivation comes from the Condevaux--Dazord--Molino proof
of the Atiyah--Guillemin--Sternberg convexity theorem in symplectic geometry.
Categories:53D20, 52B99 |
27. CJM 2010 (vol 62 pp. 1037)
Riemann Extensions of Torsion-Free Connections with Degenerate Ricci Tensor
{Correspondence} between torsion-free connections with {nilpotent skew-symmetric curvature operator} and IP Riemann
extensions is shown. Some consequences are derived in the study of
four-dimensional IP metrics and locally homogeneous affine surfaces.
Keywords:Walker metric, Riemann extension, curvature operator, projectively flat and recurrent affine connection Categories:53B30, 53C50 |
28. CJM 2009 (vol 62 pp. 52)
An Algebraic Approach to Weakly Symmetric Finsler Spaces In this paper, we introduce a new algebraic notion, weakly symmetric
Lie algebras, to give an algebraic description of an
interesting class of homogeneous Riemann--Finsler spaces, weakly symmetric
Finsler spaces. Using this new definition, we are able to give a
classification of weakly symmetric Finsler spaces with dimensions $2$
and $3$. Finally, we show that all the non-Riemannian reversible weakly
symmetric Finsler spaces we find are non-Berwaldian and with vanishing
S-curvature. This means that reversible non-Berwaldian Finsler spaces
with vanishing S-curvature may exist at large. Hence the generalized
volume comparison theorems due to Z. Shen are valid for a rather large
class of Finsler spaces.
Keywords:weakly symmetric Finsler spaces, weakly symmetric Lie algebras, Berwald spaces, S-curvature Categories:53C60, 58B20, 22E46, 22E60 |
29. CJM 2009 (vol 62 pp. 320)
Some Rigidity Results Related to MongeâAmpÃ¨re Functions The space of Monge-AmpÃ¨re functions, introduced by J. H. G. Fu, is
a space of rather rough functions in which the map $u\mapsto \operatorname{Det} D^2
u$ is well defined and weakly continuous with respect to a natural
notion of weak convergence. We prove a rigidity theorem for
Lagrangian integral currents that allows us to extend the original
definition of Monge-AmpÃ¨re functions. We also
prove that if a Monge-AmpÃ¨re function $u$ on a bounded set
$\Omega\subset\mathcal{R}^2$ satisfies the equation $\operatorname{Det} D^2 u=0$ in a
particular weak sense, then the graph of $u$ is a developable surface,
and moreover $u$ enjoys somewhat better regularity properties than an
arbitrary Monge-AmpÃ¨re function of $2$ variables.
Categories:49Q15, 53C24 |
30. CJM 2009 (vol 62 pp. 3)
On the Asymptotic Behavior of Complete KÃ¤hler Metrics of Positive Ricci Curvature Let $( X,g) $ be a complete noncompact KÃ¤hler manifold, of
dimension $n\geq2,$ with positive Ricci curvature and of standard type
(see the definition below). N. Mok proved that $X$ can be
compactified, \emph{i.e.,} $X$ is biholomorphic to a quasi-projective
variety$.$ The aim of this paper is to prove that the $L^{2}$
holomorphic sections of the line bundle $K_{X}^{-q}$ and the volume
form of the metric $g$ have no essential singularities near the
divisor at infinity. As a consequence we obtain a comparison between
the volume forms of the KÃ¤hler metric $g$ and of the Fubini--Study
metric induced on $X$. In the case of $\dim_{\mathbb{C} }X=2,$ we
establish a relation between the number of components of the divisor
$D$ and the dimension of the groups $H^{i}( \overline{X},
\Omega_{\overline{X}}^{1}( \log D) )$.
Categories:53C55, 32A10 |
31. CJM 2009 (vol 61 pp. 1357)
On a Class of Landsberg Metrics in Finsler Geometry In this paper, we study a long existing open problem on Landsberg
metrics in Finsler geometry. We consider Finsler metrics defined by a
Riemannian metric and a $1$-form on a manifold. We show that a
\emph{regular} Finsler metric in this form is Landsbergian if and only if it
is Berwaldian. We further show that there is a two-parameter family of
functions, $\phi=\phi(s)$, for which there are a Riemannian metric
$\alpha$ and a $1$-form $\beta$ on a manifold $M$ such that the scalar
function $F=\alpha \phi (\beta/\alpha)$ on $TM$ is an almost regular
Landsberg metric, but not a Berwald metric.
Categories:53B40, 53C60 |
32. 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 |
33. CJM 2009 (vol 61 pp. 740)
On Geometric Flats in the CAT(0) Realization of Coxeter Groups and Tits Buildings Given a complete CAT(0) space $X$ endowed with a geometric action of a group $\Gamma$, it is known that if
$\Gamma$ contains a free abelian group of rank $n$, then $X$ contains a geometric flat of dimension $n$. We
prove the converse of this statement in the special case where $X$ is a convex subcomplex of the CAT(0)
realization of a Coxeter group $W$, and $\Gamma$ is a subgroup of $W$. In particular a convex cocompact subgroup
of a Coxeter group is Gromov-hyperbolic if and only if it does not contain a free abelian group of rank 2. Our
result also provides an explicit control on geometric flats in the CAT(0) realization of arbitrary Tits
buildings.
Keywords:Coxeter group, flat rank, $\cat0$ space, building Categories:20F55, 51F15, 53C23, 20E42, 51E24 |
34. CJM 2009 (vol 61 pp. 721)
SubRiemannian Geometry on the Sphere $\mathbb{S}^3$ We discuss the subRiemannian
geometry induced by two noncommutative
vector fields which are left invariant
on the Lie group $\mathbb{S}^3$.
Keywords:noncommutative Lie group, quaternion group, subRiemannian geodesic, horizontal distribution, connectivity theorem, holonomic constraint Categories:53C17, 53C22, 35H20 |
35. CJM 2009 (vol 61 pp. 641)
Characterization of Parallel Isometric Immersions of Space Forms into Space Forms in the Class of Isotropic Immersions |
Characterization of Parallel Isometric Immersions of Space Forms into Space Forms in the Class of Isotropic Immersions For an isotropic submanifold $M^n\,(n\geqq3)$ of a space form
$\widetilde{M}^{n+p}(c)$ of constant sectional curvature $c$, we
show that if the mean curvature vector of $M^n$ is parallel and the
sectional curvature $K$ of $M^n$ satisfies some inequality, then
the second fundamental form of $M^n$ in $\widetilde{M}^{n+p}$ is
parallel and our manifold $M^n$ is a space form.
Keywords:space forms, parallel isometric immersions, isotropic immersions, totally umbilic, Veronese manifolds, sectional curvatures, parallel mean curvature vector Categories:53C40, 53C42 |
36. CJM 2008 (vol 60 pp. 1201)
HÃ¶lder Compactification for Some Manifolds with Pinched Negative Curvature Near Infinity We consider a complete noncompact Riemannian manifold $M$ and give
conditions on a compact submanifold $K \subset M$ so that the outward
normal exponential map off the boundary of $K$ is a diffeomorphism
onto $\MlK$. We use this to compactify $M$ and show that pinched
negative sectional curvature outside $K$ implies $M$ has a
compactification with a well-defined H\"older structure independent of
$K$. The H\"older constant depends on the ratio of the curvature
pinching. This extends and generalizes a 1985 result of Anderson and
Schoen.
Category:53C20 |
37. CJM 2008 (vol 60 pp. 822)
Maximum Principles for Subharmonic Functions Via Local Semi-Dirichlet Forms Maximum principles for subharmonic
functions in the framework of quasi-regular local semi-Dirichlet
forms admitting lower bounds are presented.
As applications, we give
weak and strong maximum principles
for (local) subsolutions of a second order elliptic
differential operator on the domain of Euclidean space under conditions on coefficients,
which partially generalize the results by Stampacchia.
Keywords:positivity preserving form, semi-Dirichlet form, Dirichlet form, subharmonic functions, superharmonic functions, harmonic functions, weak maximum principle, strong maximum principle, irreducibility, absolute continuity condition Categories:31C25, 35B50, 60J45, 35J, 53C, 58 |
38. CJM 2008 (vol 60 pp. 572)
Non-Selfadjoint Perturbations of Selfadjoint Operators in Two Dimensions IIIa. One Branching Point This is the third in a series of works devoted to spectral
asymptotics for non-selfadjoint
perturbations of selfadjoint $h$-pseudodifferential operators in dimension 2, having a
periodic classical flow. Assuming that the strength $\epsilon$
of the perturbation is in the range $h^2\ll \epsilon \ll h^{1/2}$
(and may sometimes reach even smaller values), we
get an asymptotic description of the eigenvalues in rectangles
$[-1/C,1/C]+i\epsilon [F_0-1/C,F_0+1/C]$, $C\gg 1$, when $\epsilon F_0$ is a saddle point
value of the flow average of the leading perturbation.
Keywords:non-selfadjoint, eigenvalue, periodic flow, branching singularity Categories:31C10, 35P20, 35Q40, 37J35, 37J45, 53D22, 58J40 |
39. CJM 2008 (vol 60 pp. 457)
Harmonic Coordinates on Fractals with Finitely Ramified Cell Structure We define sets with finitely ramified cell structure, which are
generalizations of post-crit8cally finite self-similar
sets introduced by Kigami and of fractafolds introduced by Strichartz. In general,
we do not assume even local self-similarity, and allow countably many cells
connected at each junction point.
In particular, we consider post-critically infinite fractals.
We prove that if Kigami's resistance form
satisfies certain assumptions, then there exists a weak Riemannian metric
such that the energy can be expressed as the integral of the norm squared
of a weak gradient with respect to an energy measure.
Furthermore, we prove that if such a set can be homeomorphically represented
in harmonic coordinates, then for smooth functions the weak gradient can be
replaced by the usual gradient.
We also prove a simple formula for the energy measure Laplacian in harmonic
coordinates.
Keywords:fractals, self-similarity, energy, resistance, Dirichlet forms, diffusions, quantum graphs, generalized Riemannian metric Categories:28A80, 31C25, 53B99, 58J65, 60J60, 60G18 |
40. CJM 2008 (vol 60 pp. 443)
On a Class of Projectively Flat Metrics with Constant Flag Curvature In this paper, we find equations that characterize locally
projectively flat Finsler metrics in the form $F = (\alpha +
\beta)^2/\alpha$, where $\alpha=\sqrt{a_{ij}y^iy^j}$ is a Riemannian
metric and $\beta= b_i y^i$ is a $1$-form. Then we completely
determine the local structure of those with constant flag curvature.
Category:53B40 |
41. 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 |
42. CJM 2007 (vol 59 pp. 981)
The Chen--Ruan Cohomology of Weighted Projective Spaces In this paper we study the Chen--Ruan cohomology ring of weighted
projective spaces. Given a weighted projective space ${\bf
P}^{n}_{q_{0}, \dots, q_{n}}$, we determine all of its twisted
sectors and the corresponding degree shifting numbers. The main
result of this paper is that the obstruction bundle over any
3\nobreakdash-multi\-sector is a direct sum of line bundles which we use to
compute the orbifold cup product. Finally we compute the
Chen--Ruan cohomology ring of weighted projective space ${\bf
P}^{5}_{1,2,2,3,3,3}$.
Keywords:Chen--Ruan cohomology, twisted sectors, toric varieties, weighted projective space, localization Categories:14N35, 53D45 |
43. CJM 2007 (vol 59 pp. 845)
Representations of the Fundamental Group of an $L$-Punctured Sphere Generated by Products of Lagrangian Involutions |
Representations of the Fundamental Group of an $L$-Punctured Sphere Generated by Products of Lagrangian Involutions In this paper, we characterize unitary representations of $\pi:=\piS$ whose
generators $u_1, \dots, u_l$ (lying in conjugacy classes fixed initially)
can be decomposed as products of two Lagrangian involutions
$u_j=\s_j\s_{j+1}$ with $\s_{l+1}=\s_1$. Our main result is that such
representations are exactly the elements of the fixed-point set of an
anti-symplectic involution defined on the moduli space
$\Mod:=\Hom_{\mathcal C}(\pi,U(n))/U(n)$. Consequently, as this fixed-point set is
non-empty, it is a Lagrangian submanifold of $\Mod$. To prove this, we use
the quasi-Hamiltonian description of the symplectic structure of $\Mod$ and
give conditions on an involution defined on a quasi-Hamiltonian $U$-space
$(M, \w, \mu\from M \to U)$ for it to induce an anti-symplectic involution on
the reduced space $M/\!/U := \mu^{-1}(\{1\})/U$.
Keywords:momentum maps, moduli spaces, Lagrangian submanifolds, anti-symplectic involutions, quasi-Hamiltonian Categories:53D20, 53D30 |
44. CJM 2006 (vol 58 pp. 600)
Geometric Study of Minkowski Differences of Plane Convex Bodies In the Euclidean plane $\mathbb{R}^{2}$, we define the Minkowski difference
$\mathcal{K}-\mathcal{L}$ of two arbitrary convex bodies $\mathcal{K}$,
$\mathcal{L}$ as a rectifiable closed curve $\mathcal{H}_{h}\subset \mathbb{R}
^{2}$ that is determined by the difference $h=h_{\mathcal{K}}-h_{\mathcal{L}
} $ of their support functions. This curve $\mathcal{H}_{h}$ is
called the
hedgehog with support function $h$. More generally, the object of hedgehog
theory is to study the Brunn--Minkowski theory in the vector space of
Minkowski differences of arbitrary convex bodies of Euclidean space $\mathbb{R}
^{n+1}$, defined as (possibly singular and self-intersecting) hypersurfaces
of $\mathbb{R}^{n+1}$. Hedgehog theory is useful for: (i)
studying convex bodies by splitting them into a sum in order to reveal their
structure; (ii) converting analytical problems into
geometrical ones by considering certain real functions as support
functions.
The purpose of this paper is to give a detailed study of plane
hedgehogs, which constitute the basis of the theory. In particular:
(i) we study their length measures and solve the extension of the
Christoffel--Minkowski problem to plane hedgehogs; (ii) we
characterize support functions of plane convex bodies among support
functions of plane hedgehogs and support functions of plane hedgehogs among
continuous functions; (iii) we study the mixed area of
hedgehogs in $\mathbb{R}^{2}$ and give an extension of the classical Minkowski
inequality (and thus of the isoperimetric inequality) to hedgehogs.
Categories:52A30, 52A10, 53A04, 52A38, 52A39, 52A40 |
45. 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 |
46. 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 |
47. CJM 2006 (vol 58 pp. 362)
Cohomology Pairings on the Symplectic Reduction of Products Let $M$ be the product of two compact Hamiltonian
$T$-spaces $X$ and $Y$. We present a formula for evaluating
integrals on the symplectic reduction of $M$ by the diagonal $T$
action. At every regular value of the moment map for $X\times Y$, the
integral is the convolution of two distributions associated to the
symplectic reductions of $X$ by $T$ and of $Y$ by $T$. Several
examples illustrate the computational strength of this relationship.
We also prove a linear analogue which can be used to find cohomology
pairings on toric orbifolds.
Category:53D20 |
48. 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 |
49. 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 |
50. CJM 2005 (vol 57 pp. 1314)
Relative Darboux Theorem for Singular Manifolds and Local Contact Algebra In 1999 V. Arnol'd introduced the local contact algebra: studying the
problem of classification of singular curves in a contact space, he
showed the existence of the ghost of the contact structure (invariants
which are not related to the induced structure on the curve). Our
main result implies that the only reason for existence of the local
contact algebra and the ghost is the difference between the geometric
and (defined in this paper) algebraic restriction of a $1$-form to a
singular submanifold. We prove that a germ of any subset $N$ of a
contact manifold is well defined, up to contactomorphisms, by the
algebraic restriction to $N$ of the contact structure. This is a
generalization of the Darboux-Givental' theorem for smooth
submanifolds of a contact manifold. Studying the difference between
the geometric and the algebraic restrictions gives a powerful tool for
classification of stratified submanifolds of a contact manifold. This
is illustrated by complete solution of three classification problems,
including a simple explanation of V.~Arnold's results and further
classification results for singular curves in a contact space. We
also prove several results on the external geometry of a singular
submanifold $N$ in terms of the algebraic restriction of the contact
structure to $N$. In particular, the algebraic restriction is zero if
and only if $N$ is contained in a smooth Legendrian submanifold of
$M$.
Keywords:contact manifold, local contact algebra,, relative Darboux theorem, integral curves Categories:53D10, 14B05, 58K50 |