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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 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
Category:53C30

27. 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

28. 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

29. 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

30. 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

31. 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

32. 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

33. 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

34. 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

35. 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

36. 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

37. 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

38. 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

39. 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

40. 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

41. 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

42. 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

43. 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 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

44. CJM 1997 (vol 49 pp. 696)

Charitos, Charalambos; Tsapogas, Georgios
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

45. 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 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

46. CJM 1997 (vol 49 pp. 359)

Sawyer, P.
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
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