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51. CMB 2006 (vol 49 pp. 152)

Yun, Jong-Gug
 Comparison Geometry With\\$L^1$-Norms of Ricci Curvature We investigate the geometry of manifolds with bounded Ricci curvature in $L^1$-sense. In particular, we generalize the classical volume comparison theorem to our situation and obtain a generalized sphere theorem. Keywords:Mean curvature, Ricci curvatureCategory:53C20

52. CMB 2006 (vol 49 pp. 36)

Daskalopoulos, Georgios D.; Wentworth, Richard A.
 Holomorphic Frames for Weakly Converging Holomorphic Vector Bundles Using a modification of Webster's proof of the Newlander--Nirenberg theorem, it is shown that, for a weakly convergent sequence of integrable unitary connections on a complex vector bundle over a complex manifold, there is a subsequence of local holomorphic frames that converges strongly in an appropriate Holder class. Categories:57M50, 58E20, 53C24

53. CMB 2005 (vol 48 pp. 561)

Foth, Philip
 A Note on Lagrangian Loci of Quotients We study Hamiltonian actions of compact groups in the presence of compatible involutions. We show that the Lagrangian fixed point set on the symplectically reduced space is isomorphic to the disjoint union of the involutively reduced spaces corresponding to involutions on the group strongly inner to the given one. Our techniques imply that the solution to the eigenvalues of a sum problem for a given real form can be reduced to the quasi-split real form in the same inner class. We also consider invariant quotients with respect to the corresponding real form of the complexified group. Keywords:Quotients, involutions, real forms, Lagrangian lociCategory:53D20

54. CMB 2005 (vol 48 pp. 112)

Mo, Xiaohuan; Shen, Zhongmin
 On Negatively Curved Finsler Manifolds of Scalar Curvature In this paper, we prove a global rigidity theorem for negatively curved Finsler metrics on a compact manifold of dimension $n \geq 3$. We show that for such a Finsler manifold, if the flag curvature is a scalar function on the tangent bundle, then the Finsler metric is of Randers type. We also study the case when the Finsler metric is locally projectively flat Category:53C60

55. CMB 2004 (vol 47 pp. 624)

Zhang, Xi
 A Compactness Theorem for Yang-Mills Connections In this paper, we consider Yang-Mills connections on a vector bundle $E$ over a compact Riemannian manifold $M$ of dimension $m> 4$, and we show that any set of Yang-Mills connections with the uniformly bounded $L^{\frac{m}{2}}$-norm of curvature is compact in $C^{\infty}$ topology. Keywords:Yang-Mills connection, vector bundle, gauge transformationCategories:58E20, 53C21

56. CMB 2004 (vol 47 pp. 492)

Boumuki, Nobutaka
 Isotropic Immersions with Low Codimension of Complex Space Forms into Real Space Forms The main purpose of this paper is to determine isotropic immersions of complex space forms into real space forms with low codimension. This is an improvement of a result of S. Maeda. Categories:53B25, 53C235

57. CMB 2004 (vol 47 pp. 354)

Fawaz, Amine
 An Integral Formula on Seifert Bundles We prove an integral formula on closed oriented manifolds equipped with a codimension two foliation whose leaves are compact. Categories:53C12, 53C15.

58. CMB 2004 (vol 47 pp. 314)

Yun, Jong-Gug
 Mean Curvature Comparison with $L^1$-norms of Ricci Curvature We prove an analogue of mean curvature comparison theorem in the case where the Ricci curvature below a positive constant is small in $L^1$-norm. Keywords:mean curvature, Ricci curvatureCategory:53C20

59. CMB 2003 (vol 46 pp. 617)

Pak, Hong Kyung
 On Harmonic Theory in Flows Recently [8], a harmonic theory was developed for a compact contact manifold from the viewpoint of the transversal geometry of contact flow. A contact flow is a typical example of geodesible flow. As a natural generalization of the contact flow, the present paper develops a harmonic theory for various flows on compact manifolds. We introduce the notions of $H$-harmonic and $H^*$-harmonic spaces associated to a H\"ormander flow. We also introduce the notions of basic harmonic spaces associated to a weak basic flow. One of our main results is to show that in the special case of isometric flow these harmonic spaces are isomorphic to the cohomology spaces of certain complexes. Moreover, we find an obstruction for a geodesible flow to be isometric. Keywords:contact structure, geodesible flow, isometric flow, basic cohomologyCategories:53C20, 57R30

60. CMB 2003 (vol 46 pp. 277)

Rochon, Frédéric
 Rigidity of Hamiltonian Actions This paper studies the following question: Given an $\omega'$-symplectic action of a Lie group on a manifold $M$ which coincides, as a smooth action, with a Hamiltonian $\omega$-action, when is this action a Hamiltonian $\omega'$-action? Using a result of Morse-Bott theory presented in Section~2, we show in Section~3 of this paper that such an action is in fact a Hamiltonian $\omega'$-action, provided that $M$ is compact and that the Lie group is compact and connected. This result was first proved by Lalonde-McDuff-Polterovich in 1999 as a consequence of a more general theory that made use of hard geometric analysis. In this paper, we prove it using classical methods only. Categories:53D05, 37J25

61. CMB 2003 (vol 46 pp. 130)

Petersen, Peter; Wilhelm, Frederick
 On Frankel's Theorem In this paper we show that two minimal hypersurfaces in a manifold with positive Ricci curvature must intersect. This is then generalized to show that in manifolds with positive Ricci curvature in the integral sense two minimal hypersurfaces must be close to each other. We also show what happens if a manifold with nonnegative Ricci curvature admits two nonintersecting minimal hypersurfaces. Keywords:Frankel's TheoremCategory:53C20

62. CMB 2002 (vol 45 pp. 378)

Fernández-López, Manuel; García-Río, Eduardo; Kupeli, Demir N.
 The Local MÃ¶bius Equation and Decomposition Theorems in Riemannian Geometry A partial differential equation, the local M\"obius equation, is introduced in Riemannian geometry which completely characterizes the local twisted product structure of a Riemannian manifold. Also the characterizations of warped product and product structures of Riemannian manifolds are made by the local M\"obius equation and an additional partial differential equation. Keywords:submersion, MÃ¶bius equation, twisted product, warped product, product Riemannian manifoldsCategories:53C12, 58J99

63. CMB 2002 (vol 45 pp. 161)

Ardizzone, Lucia; Grimaldi, Renata; Pansu, Pierre
 Sur les singularitÃ©s de la fonction croissance d'une variÃ©tÃ© non simplement connexe Si $M$ est une vari\'et\'e de dimension $n$, compacte non simplement connexe, on caract\'erise les m\'etriques riemanniennes sur $M$ dont la fonction croissance a exactement deux singularit\'es. Keywords:fonction croissance, singularitÃ©s, fonction de Morse, CutlocusCategory:53B20

64. CMB 2002 (vol 45 pp. 232)

Ji, Min; Shen, Zhongmin
 On Strongly Convex Indicatrices in Minkowski Geometry The geometry of indicatrices is the foundation of Minkowski geometry. A strongly convex indicatrix in a vector space is a strongly convex hypersurface. It admits a Riemannian metric and has a distinguished invariant---(Cartan) torsion. We prove the existence of non-trivial strongly convex indicatrices with vanishing mean torsion and discuss the relationship between the mean torsion and the Riemannian curvature tensor for indicatrices of Randers type. Categories:46B20, 53C21, 53A55, 52A20, 53B40, 53A35

65. CMB 2001 (vol 44 pp. 408)

Falbel, E.
 Finite Groups Generated by Involutions on Lagrangian Planes of $\mathbf{C}^2$ We classify finite subgroups of $\SO(4)$ generated by anti-unitary involutions. They correspond to involutions fixing pointwise a Lagrangian plane. Explicit descriptions of the finite groups and the configurations of Lagrangian planes are obtained. Categories:22E40, 53D99

66. CMB 2001 (vol 44 pp. 376)

Zhang, Xi
 A Note on $p$-Harmonic $1$-Forms on Complete Manifolds In this paper we prove that there is no nontrivial $L^{q}$-integrably $p$-harmonic $1$-form on a complete manifold with nonnegatively Ricci curvature $(0 Keywords:$p$-harmonic,$1$-form, complete manifold, Sobolev inequalityCategories:58E20, 53C21 67. CMB 2001 (vol 44 pp. 129) Currás-Bosch, Carlos  LinÃ©arisation symplectique en dimension 2 In this paper the germ of neighborhood of a compact leaf in a Lagrangian foliation is symplectically classified when the compact leaf is$\bT^2$, the affine structure induced by the Lagrangian foliation on the leaf is complete, and the holonomy of$\bT^2$in the foliation linearizes. The germ of neighborhood is classified by a function, depending on one transverse coordinate, this function is related to the affine structure of the nearly compact leaves. Keywords:symplectic manifold, Lagrangian foliation, affine connectionCategories:53C12, 58F05 68. CMB 2001 (vol 44 pp. 70) Lempert, László; Szőke, Róbert  The Tangent Bundle of an Almost Complex Manifold Motivated by deformation theory of holomorphic maps between almost complex manifolds we endow, in a natural way, the tangent bundle of an almost complex manifold with an almost complex structure. We describe various properties of this structure. Category:53C15 69. CMB 2001 (vol 44 pp. 36) Kapovich, Michael; Millson, John J.  Quantization of Bending Deformations of Polygons In$\mathbb{E}^3$, Hypergeometric Integrals and the Gassner Representation The Hamiltonian potentials of the bending deformations of$n$-gons in$\E^3$studied in \cite{KM} and \cite{Kl} give rise to a Hamiltonian action of the Malcev Lie algebra$\p_n$of the pure braid group$P_n$on the moduli space$M_r$of$n$-gon linkages with the side-lengths$r= (r_1,\dots, r_n)$in$\E^3$. If$e\in M_r$is a singular point we may linearize the vector fields in$\p_n$at$e$. This linearization yields a flat connection$\nabla$on the space$\C^n_*$of$n$distinct points on$\C$. We show that the monodromy of$\nabla$is the dual of a quotient of a specialized reduced Gassner representation. Categories:53D30, 53D50 70. CMB 2000 (vol 43 pp. 427) Ivey, Thomas A.  Helices, Hasimoto Surfaces and BÃ¤cklund Transformations Travelling wave solutions to the vortex filament flow generated by elastica produce surfaces in$\R^3$that carry mutually orthogonal foliations by geodesics and by helices. These surfaces are classified in the special cases where the helices are all congruent or are all generated by a single screw motion. The first case yields a new characterization for the B\"acklund transformation for constant torsion curves in$\R^3$, previously derived from the well-known transformation for pseudospherical surfaces. A similar investigation for surfaces in$H^3$or$S^3$leads to a new transformation for constant torsion curves in those spaces that is also derived from pseudospherical surfaces. Keywords:surfaces, filament flow, BÃ¤cklund transformationsCategories:53A05, 58F37, 52C42, 58A15 71. CMB 2000 (vol 43 pp. 440) Koufogiorgos, Themis; Tsichlias, Charalambos  On the Existence of a New Class of Contact Metric Manifolds A new class of 3-dimensional contact metric manifolds is found. Moreover it is proved that there are no such manifolds in dimensions greater than 3. Keywords:contact metric manifolds, generalized$(\kappa,\mu)$-contact metric manifoldsCategories:53C25, 53C15 72. CMB 2000 (vol 43 pp. 343) Hughes, Bruce; Taylor, Larry; Williams, Bruce  Controlled Homeomorphisms Over Nonpositively Curved Manifolds We obtain a homotopy splitting of the forget control map for controlled homeomorphisms over closed manifolds of nonpositive curvature. Keywords:controlled topology, controlled homeomorphism, nonpositive curvature, Novikov conjecturesCategories:57N15, 53C20, 55R65, 57N37 73. CMB 2000 (vol 43 pp. 74) Kimura, Makoto; Maeda, Sadahiro  Geometric Meaning of Isoparametric Hypersurfaces in a Real Space Form We shall provide a characterization of all isoparametric hypersurfaces$M$'s in a real space form$\tilde{M}(c)$by observing the extrinsic Wshape of geodesics of$M$in the ambient manifold$\tilde{M}(c)$. Categories:53C35, 53C20, 53C22 74. CMB 1999 (vol 42 pp. 486) Sawyer, P.  Spherical Functions on$\SO_0(p,q)/\SO(p)\times \SO(q)$An integral formula is derived for the spherical functions on the symmetric space$G/K=\break \SO_0(p,q)/\SO(p)\times \SO(q)$. This formula allows us to state some results about the analytic continuation of the spherical functions to a tubular neighbourhood of the subalgebra$\a$of the abelian part in the decomposition$G=KAK$. The corresponding result is then obtained for the heat kernel of the symmetric space$\SO_0(p,q)/\SO (p)\times\SO (q)$using the Plancherel formula. In the Conclusion, we discuss how this analytic continuation can be a helpful tool to study the growth of the heat kernel. Categories:33C55, 17B20, 53C35 75. CMB 1999 (vol 42 pp. 214) Paeng, Seong-Hun; Yun, Jong-Gug  Conjugate Radius and Sphere Theorem Bessa [Be] proved that for given$n$and$i_0$, there exists an$\varepsilon(n,i_0)>0$depending on$n,i_0$such that if$M$admits a metric$g$satisfying$\Ric_{(M,g)} \ge n-1$,$\inj_{(M,g)} \ge i_0>0$and$\diam_{(M,g)} \ge \pi-\varepsilon$, then$M\$ is diffeomorphic to the standard sphere. In this note, we improve this result by replacing a lower bound on the injectivity radius with a lower bound of the conjugate radius. Keywords:Ricci curvature, conjugate radiusCategories:53C20, 53C21
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