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51. CMB 2007 (vol 50 pp. 347)

Pérez, Juan de Dios; Santos, Florentino G.; Suh, Young Jin
 Real Hypersurfaces in Complex Projective Space Whose Structure Jacobi Operator Is of Codazzi Type We prove the non existence of real hypersurfaces in complex projective space whose structure Jacobi operator is of Codazzi type. Categories:53C15, 53B25

52. CMB 2007 (vol 50 pp. 474)

Zhou, Jiazu
 On Willmore's Inequality for Submanifolds Let $M$ be an $m$ dimensional submanifold in the Euclidean space ${\mathbf R}^n$ and $H$ be the mean curvature of $M$. We obtain some low geometric estimates of the total square mean curvature $\int_M H^2 d\sigma$. The low bounds are geometric invariants involving the volume of $M$, the total scalar curvature of $M$, the Euler characteristic and the circumscribed ball of $M$. Keywords:submanifold, mean curvature, kinematic formul, scalar curvatureCategories:52A22, 53C65, 51C16

53. CMB 2007 (vol 50 pp. 365)

Godinho, Leonor
 Equivariant Cohomology of $S^{1}$-Actions on $4$-Manifolds Let $M$ be a symplectic $4$-dimensional manifold equipped with a Hamiltonian circle action with isolated fixed points. We describe a method for computing its integral equivariant cohomology in terms of fixed point data. We give some examples of these computations. Categories:53D20, 55N91, 57S15

54. CMB 2007 (vol 50 pp. 321)

Blair, David E.
 On Lagrangian Catenoids Recently I. Castro and F. Urbano introduced the Lagrangian catenoid. Topologically, it is $\mathbb R\times S^{n-1}$ and its induced metric is conformally flat, but not cylindrical. Their result is that if a Lagrangian minimal submanifold in ${\mathbb C}^n$ is foliated by round $(n-1)$-spheres, it is congruent to a Lagrangian catenoid. Here we study the question of conformally flat, minimal, Lagrangian submanifolds in ${\mathbb C}^n$. The general problem is formidable, but we first show that such a submanifold resembles a Lagrangian catenoid in that its Schouten tensor has an eigenvalue of multiplicity one. Then, restricting to the case of at most two eigenvalues, we show that the submanifold is either flat and totally geodesic or is homothetic to (a piece of) the Lagrangian catenoid. Categories:53C42, 53D12

55. CMB 2007 (vol 50 pp. 97)

Kim, In-Bae; Kim, Ki Hyun; Sohn, Woon Ha
 Characterizations of Real Hypersurfaces in a Complex Space Form We study a real hypersurface $M$ in a complex space form $\mn$, $c \neq 0$, whose shape operator and structure tensor commute each other on the holomorphic distribution of $M$. Categories:53C40, 53C15

56. CMB 2007 (vol 50 pp. 113)

Li, ZhenYang; Zhang, Xi
 Hermitian Harmonic Maps into Convex Balls In this paper, we consider Hermitian harmonic maps from Hermitian manifolds into convex balls. We prove that there exist no non-trivial Hermitian harmonic maps from closed Hermitian manifolds into convex balls, and we use the heat flow method to solve the Dirichlet problem for Hermitian harmonic maps when the domain is a compact Hermitian manifold with non-empty boundary. Keywords:Hermitian harmonic map, Hermitian manifold, convex ballCategories:58E15, 53C07

57. CMB 2007 (vol 50 pp. 24)

Brown, Nathan; Finck, Rachel; Spencer, Matthew; Tapp, Kristopher; Wu, Zhongtao
 Invariant Metrics with Nonnegative Curvature on Compact Lie Groups We classify the left-invariant metrics with nonnegative sectional curvature on $\SO(3)$ and $U(2)$. Category:53C20

58. CMB 2006 (vol 49 pp. 321)

Balser, Andreas
 Polygons with Prescribed Gauss Map in Hadamard Spaces and Euclidean Buildings We show that given a stable weighted configuration on the asymptotic boundary of a locally compact Hadamard space, there is a polygon with Gauss map prescribed by the given weighted configuration. Moreover, the same result holds for semistable configurations on arbitrary Euclidean buildings. Keywords:Euclidean buildings, Hadamard spaces, polygonsCategory:53C20

59. CMB 2006 (vol 49 pp. 226)

Engman, Martin
 The Spectrum and Isometric Embeddings of Surfaces of Revolution A sharp upper bound on the first $S^{1}$ invariant eigenvalue of the Laplacian for $S^1$ invariant metrics on $S^2$ is used to find obstructions to the existence of $S^1$ equivariant isometric embeddings of such metrics in $(\R^3,\can)$. As a corollary we prove: If the first four distinct eigenvalues have even multiplicities then the metric cannot be equivariantly, isometrically embedded in $(\R^3,\can)$. This leads to generalizations of some classical results in the theory of surfaces. Categories:58J50, 58J53, 53C20, 35P15

60. CMB 2006 (vol 49 pp. 134)

Suh, Young Jin
 Real Hypersurfaces in Complex Two-Plane Grassmannians with Vanishing Lie Derivative In this paper we give a characterization of real hypersurfaces of type $A$ in a complex two-plane Grassmannian $G_2(\mathbb{C}^{m+2})$ which are tubes over totally geodesic $G_2(\mathbb{C}^{m+1})$ in $G_2(\mathbb{C}^{m+2})$ in terms of the {\it vanishing Lie derivative\/} of the shape operator $A$ along the direction of the Reeb vector field $\xi$. Categories:53C40, 53C15

61. 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

62. 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

63. 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

64. 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

65. 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

66. 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

67. 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.

68. 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

69. 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

70. 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

71. 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

72. 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

73. 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

74. 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

75. 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
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