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26. CMB 2012 (vol 57 pp. 12)

Aribi, Amine; Dragomir, Sorin; El Soufi, Ahmad
 On the Continuity of the Eigenvalues of a Sublaplacian We study the behavior of the eigenvalues of a sublaplacian $\Delta_b$ on a compact strictly pseudoconvex CR manifold $M$, as functions on the set ${\mathcal P}_+$ of positively oriented contact forms on $M$ by endowing ${\mathcal P}_+$ with a natural metric topology. Keywords:CR manifold, contact form, sublaplacian, Fefferman metricCategories:32V20, 53C56

27. CMB 2011 (vol 56 pp. 306)

Pérez, Juan de Dios; Suh, Young Jin
 Real Hypersurfaces in Complex Projective Space Whose Structure Jacobi Operator is Lie $\mathbb{D}$-parallel We prove the non-existence of real hypersurfaces in complex projective space whose structure Jacobi operator is Lie $\mathbb{D}$-parallel and satisfies a further condition. Keywords:complex projective space, real hypersurface, structure Jacobi operatorCategories:53C15, 53C40

28. CMB 2011 (vol 56 pp. 184)

Shen, Zhongmin
 On Some Non-Riemannian Quantities in Finsler Geometry In this paper we study several non-Riemannian quantities in Finsler geometry. These non-Riemannian quantities play an important role in understanding the geometric properties of Finsler metrics. In particular, we study a new non-Riemannian quantity defined by the S-curvature. We show some relationships among the flag curvature, the S-curvature, and the new non-Riemannian quantity. Keywords:Finsler metric, S-curvature, non-Riemannian quantityCategories:53C60, 53B40

29. CMB 2011 (vol 56 pp. 615)

Sevim, Esra Sengelen; Shen, Zhongmin
 Randers Metrics of Constant Scalar Curvature Randers metrics are a special class of Finsler metrics. Every Randers metric can be expressed in terms of a Riemannian metric and a vector field via Zermelo navigation. In this paper, we show that a Randers metric has constant scalar curvature if the Riemannian metric has constant scalar curvature and the vector field is homothetic. Keywords:Randers metrics, scalar curvature, S-curvatureCategories:53C60, 53B40

30. CMB 2011 (vol 56 pp. 127)

Li, Junfang
 Evolution of Eigenvalues along Rescaled Ricci Flow In this paper, we discuss monotonicity formulae of various entropy functionals under various rescaled versions of Ricci flow. As an application, we prove that the lowest eigenvalue of a family of geometric operators $-4\Delta + kR$ is monotonic along the normalized Ricci flow for all $k\ge 1$ provided the initial manifold has nonpositive total scalar curvature. Keywords:monotonicity formulas, Ricci flowCategories:58C40, 53C44

31. CMB 2011 (vol 55 pp. 870)

Wang, Hui; Deng, Shaoqiang
 Left Invariant Einstein-Randers Metrics on Compact Lie Groups In this paper we study left invariant Einstein-Randers metrics on compact Lie groups. First, we give a method to construct left invariant non-Riemannian Einstein-Randers metrics on a compact Lie group, using the Zermelo navigation data. Then we prove that this gives a complete classification of left invariant Einstein-Randers metrics on compact simple Lie groups with the underlying Riemannian metric naturally reductive. Further, we completely determine the identity component of the group of isometries for this type of metrics on simple groups. Finally, we study some geometric properties of such metrics. In particular, we give the formulae of geodesics and flag curvature of such metrics. Keywords:Einstein-Randers metric, compact Lie groups, geodesic, flag curvatureCategories:17B20, 22E46, 53C12

32. CMB 2011 (vol 56 pp. 173)

Sahin, Bayram
 Semi-invariant Submersions from Almost Hermitian Manifolds We introduce semi-invariant Riemannian submersions from almost Hermitian manifolds onto Riemannian manifolds. We give examples, investigate the geometry of foliations that arise from the definition of a Riemannian submersion, and find necessary sufficient conditions for total manifold to be a locally product Riemannian manifold. We also find necessary and sufficient conditions for a semi-invariant submersion to be totally geodesic. Moreover, we obtain a classification for semi-invariant submersions with totally umbilical fibers and show that such submersions put some restrictions on total manifolds. Keywords:Riemannian submersion, Hermitian manifold, anti-invariant Riemannian submersion, semi-invariant submersionCategories:53B20, 53C43

33. CMB 2011 (vol 55 pp. 632)

Pigola, S.; Rimoldi, M.
 Characterizations of Model Manifolds by Means of Certain Differential Systems We prove metric rigidity for complete manifolds supporting solutions of certain second order differential systems, thus extending classical works on a characterization of space-forms. Along the way, we also discover new characterizations of space-forms. We next generalize results concerning metric rigidity via equations involving vector fields. Keywords:metric rigidity, model manifolds, Obata's type theoremsCategory:53C20

34. CMB 2011 (vol 56 pp. 116)

Krepski, Derek
 Central Extensions of Loop Groups and Obstruction to Pre-Quantization An explicit construction of a pre-quantum line bundle for the moduli space of flat $G$-bundles over a Riemann surface is given, where $G$ is any non-simply connected compact simple Lie group. This work helps to explain a curious coincidence previously observed between Toledano Laredo's work classifying central extensions of loop groups $LG$ and the author's previous work on the obstruction to pre-quantization of the moduli space of flat $G$-bundles. Keywords:loop group, central extension, prequantizationCategories:53D, 22E

35. CMB 2011 (vol 55 pp. 723)

Gigli, Nicola; Ohta, Shin-Ichi
 First Variation Formula in Wasserstein Spaces over Compact Alexandrov Spaces We extend results proved by the second author (Amer. J. Math., 2009) for nonnegatively curved Alexandrov spaces to general compact Alexandrov spaces $X$ with curvature bounded below. The gradient flow of a geodesically convex functional on the quadratic Wasserstein space $(\mathcal P(X),W_2)$ satisfies the evolution variational inequality. Moreover, the gradient flow enjoys uniqueness and contractivity. These results are obtained by proving a first variation formula for the Wasserstein distance. Keywords:Alexandrov spaces, Wasserstein spaces, first variation formula, gradient flowCategories:53C23, 28A35, 49Q20, 58A35

36. CMB 2011 (vol 55 pp. 663)

Zhou, Chunqin
 An Onofri-type Inequality on the Sphere with Two Conical Singularities In this paper, we give a new proof of the Onofri-type inequality \begin{equation*} \int_S e^{2u} \,ds^2 \leq 4\pi(\beta+1) \exp \biggl\{ \frac{1}{4\pi(\beta+1)} \int_S |\nabla u|^2 \,ds^2 + \frac{1}{2\pi(\beta+1)} \int_S u \,ds^2 \biggr\} \end{equation*} on the sphere $S$ with Gaussian curvature $1$ and with conical singularities divisor $\mathcal A = \beta\cdot p_1 + \beta \cdot p_2$ for $\beta\in (-1,0)$; here $p_1$ and $p_2$ are antipodal. Categories:53C21, 35J61, 53A30

37. CMB 2011 (vol 56 pp. 44)

Biswas, Indranil; Dey, Arijit
 Polystable Parabolic Principal $G$-Bundles and Hermitian-Einstein Connections We show that there is a bijective correspondence between the polystable parabolic principal $G$-bundles and solutions of the Hermitian-Einstein equation. Keywords:ramified principal bundle, parabolic principal bundle, Hitchin-Kobayashi correspondence, polystabilityCategories:32L04, 53C07

38. CMB 2011 (vol 55 pp. 611)

 Chen Inequalities for Submanifolds of Real Space Forms with a Semi-Symmetric Non-Metric Connection In this paper we prove Chen inequalities for submanifolds of real space forms endowed with a semi-symmetric non-metric connection, i.e., relations between the mean curvature associated with a semi-symmetric non-metric connection, scalar and sectional curvatures, Ricci curvatures and the sectional curvature of the ambient space. The equality cases are considered. Keywords:real space form, semi-symmetric non-metric connection, Ricci curvatureCategories:53C40, 53B05, 53B15

39. CMB 2011 (vol 55 pp. 108)

Guler, Dincer
 On Segre Forms of Positive Vector Bundles The goal of this note is to prove that the signed Segre forms of Griffiths' positive vector bundles are positive. Categories:53C55, 32L05

40. CMB 2011 (vol 55 pp. 474)

Chen, Bin; Zhao, Lili
 A Note on Randers Metrics of Scalar Flag Curvature Some families of Randers metrics of scalar flag curvature are studied in this paper. Explicit examples that are neither locally projectively flat nor of isotropic $S$-curvature are given. Certain Randers metrics with Einstein $\alpha$ are considered and proved to be complex. Three dimensional Randers manifolds, with $\alpha$ having constant scalar curvature, are studied. Keywords:Randers metrics, scalar flag curvatureCategories:53B40, 53C60

41. CMB 2011 (vol 55 pp. 329)

Kamiya, Shigeyasu; Parker, John R.; Thompson, James M.
 Non-Discrete Complex Hyperbolic Triangle Groups of Type $(n,n, \infty;k)$ A complex hyperbolic triangle group is a group generated by three involutions fixing complex lines in complex hyperbolic space. Our purpose in this paper is to improve a previous result and to discuss discreteness of complex hyperbolic triangle groups of type $(n,n,\infty;k)$. Keywords:complex hyperbolic triangle groupCategories:51M10, 32M15, 53C55, 53C35

42. CMB 2011 (vol 55 pp. 138)

Li, Benling; Shen, Zhongmin
 Projectively Flat Fourth Root Finsler Metrics In this paper, we study locally projectively flat fourth root Finsler metrics and their generalized metrics. We prove that if they are irreducible, then they must be locally Minkowskian. Keywords:projectively flat, Finsler metric, fourth root Finsler metricCategory:53B40

43. CMB 2011 (vol 55 pp. 114)

Kon, S. H.; Loo, Tee-How
 On Characterizations of Real Hypersurfaces in a Complex Space Form with $\eta$-Parallel Shape Operator In this paper we study real hypersurfaces in a non-flat complex space form with $\eta$-parallel shape operator. Several partial characterizations of these real hypersurfaces are obtained. Keywords:complex space form, Hopf hypersurfaces, ruled real hypersurfaces, $\eta$-parallel shape operatorCategories:53C40, 53C15

44. CMB 2011 (vol 54 pp. 716)

Okassa, Eugène
 Symplectic Lie-Rinehart-Jacobi Algebras and Contact Manifolds We give a characterization of contact manifolds in terms of symplectic Lie-Rinehart-Jacobi algebras. We also give a sufficient condition for a Jacobi manifold to be a contact manifold. Keywords:Lie-Rinehart algebras, differential operators, Jacobi manifolds, symplectic manifolds, contact manifoldsCategories:13N05, 53D05, 53D10

45. CMB 2011 (vol 54 pp. 422)

Pérez, Juan de Dios; Suh, Young Jin
 Two Conditions on the Structure Jacobi Operator for Real Hypersurfaces in Complex Projective Space We classify real hypersurfaces in complex projective space whose structure Jacobi operator satisfies two conditions at the same time. Keywords:complex projective space, real hypersurface, structure Jacobi operator, two conditionsCategories:53C15, 53B25

46. CMB 2010 (vol 53 pp. 684)

Proctor, Emily; Stanhope, Elizabeth
 An Isospectral Deformation on an Infranil-Orbifold We construct a Laplace isospectral deformation of metrics on an orbifold quotient of a nilmanifold. Each orbifold in the deformation contains singular points with order two isotropy. Isospectrality is obtained by modifying a generalization of Sunada's theorem due to DeTurck and Gordon. Keywords:spectral geometry, global Riemannian geometry, orbifold, nilmanifoldCategories:58J53, 53C20

47. CMB 2010 (vol 53 pp. 564)

Watanabe, Yoshiyuki; Suh, Young Jin
 On $6$-Dimensional Nearly KÃ¤hler Manifolds In this paper we give a sufficient condition for a complete, simply connected, and strict nearly KÃ¤hler manifold of dimension 6 to be a homogeneous nearly KÃ¤hler manifold. This result was announced in a previous paper by the first author. Keywords:Nearly KÃ¤hler manifold, 6-dimension, Homogeneous, The 1st Chern Class, Einstein manifoldsCategories:53C40, 53C15

48. CMB 2010 (vol 53 pp. 516)

Maurmann, Quinn; Engelstein, Max; Marcuccio, Anthony; Pritchard, Taryn
 Asymptotics of Perimeter-Minimizing Partitions We prove that the least perimeter $P(n)$ of a partition of a smooth, compact Riemannian surface into $n$ regions of equal area $A$ is asymptotic to $n/2$ times the perimeter of a planar regular hexagon of area $A$. Along the way, we derive tighter estimates for flat tori, Klein bottles, truncated cylinders, and MÃ¶bius bands. Category:53C42

49. CMB 2010 (vol 53 pp. 412)

Calvaruso, G.
 Einstein-Like Lorentz Metrics and Three-Dimensional Curvature Homogeneity of Order One We completely classify three-dimensional Lorentz manifolds, curvature homogeneous up to order one, equipped with Einstein-like metrics. New examples arise with respect to both homogeneous examples and three-dimensional Lorentz manifolds admitting a degenerate parallel null line field. Keywords:Lorentz manifolds, curvature homogeneity, Einstein-like metricsCategories:53C50, 53C20, 53C30

50. CMB 2009 (vol 53 pp. 206)

Atçeken, Mehmet
 Semi-Slant Submanifolds of an Almost Paracontact Metric Manifold In this paper, we define and study the geometry of semi-slant submanifolds of an almost paracontact metric manifold. We give some characterizations for a submanifold to be semi-slant submanifold to be semi-slant product and obtain integrability conditions for the distributions involved in the definition of a semi-slant submanifold. Keywords:paracontact metric manifold, slant distribution, semi-slant submanifold, semi-slant productCategories:53C15, 53C25, 53C40
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