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Search: MSC category 53C ( Global differential geometry [See also 51H25, 58-XX; for related bundle theory, see 55Rxx, 57Rxx] )

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26. CJM 2009 (vol 61 pp. 1201)

Arvanitoyeorgos, Andreas; Dzhepko, V. V.; Nikonorov, Yu. G.
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

27. CJM 2009 (vol 61 pp. 740)

Caprace, Pierre-Emmanuel; Haglund, Frédéric
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

28. CJM 2009 (vol 61 pp. 721)

Calin, Ovidiu; Chang, Der-Chen; Markina, Irina
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

29. CJM 2009 (vol 61 pp. 641)

Maeda, Sadahiro; Udagawa, Seiichi
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

30. CJM 2008 (vol 60 pp. 1201)

Bahuaud, Eric; Marsh, Tracey
Hölder 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.


31. CJM 2008 (vol 60 pp. 822)

Kuwae, Kazuhiro
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

32. CJM 2007 (vol 59 pp. 1245)

Chen, Qun; Zhou, Zhen-Rong
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

33. CJM 2006 (vol 58 pp. 262)

Biswas, Indranil
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

34. 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

35. CJM 2006 (vol 58 pp. 282)

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

36. 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

37. 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

38. 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

39. CJM 2005 (vol 57 pp. 871)

Zhang, Xi
Hermitian Yang-_Mills--Higgs Metrics on\\Complete Kähler 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, Kähler manifolds
Categories:58E15, 53C07

40. 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 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

41. 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 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

42. 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

43. 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

44. 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

45. 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 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

46. CJM 2002 (vol 54 pp. 449)

Akrout, H.
Théorème de Vorono\"\i\ dans les espaces symétriques
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:réseaux, théorème de Vorono\"\i, espaces symétriques
Categories:11H06, 53C35

47. CJM 2001 (vol 53 pp. 780)

Nicolaescu, Liviu I.
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

48. CJM 2000 (vol 52 pp. 757)

Hanani, Abdellah
Le problème de Neumann pour certaines équations du type de Monge-Ampère sur une variété 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, équations de Monge-Ampère, problème de Neumann, estimées a priori, méthode de continuité
Categories:35J60, 53C55, 58G30

49. CJM 1998 (vol 50 pp. 1298)

Milson, Robert
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

50. CJM 1997 (vol 49 pp. 1162)

Ku, Hsu-Tung; Ku, Mei-Chin; Zhang, Xin-Min
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
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