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1. CMB Online first

Martinez-Maure, Yves
Plane Lorentzian and Fuchsian Hedgehogs
Parts of the Brunn-Minkowski theory can be extended to hedgehogs, which are envelopes of families of affine hyperplanes parametrized by their Gauss map. F. Fillastre introduced Fuchsian convex bodies, which are the closed convex sets of Lorentz-Minkowski space that are globally invariant under the action of a Fuchsian group. In this paper, we undertake a study of plane Lorentzian and Fuchsian hedgehogs. In particular, we prove the Fuchsian analogues of classical geometrical inequalities (analogues which are reversed as compared to classical ones).

Keywords:Fuchsian and Lorentzian hedgehogs, evolute, duality, convolution, reversed isoperimetric inequality, reversed Bonnesen inequality
Categories:52A40, 52A55, 53A04, 53B30

2. CMB Online first

da Silva, Rosângela Maria; Tenenblat, Keti
Helicoidal Minimal Surfaces in a Finsler Space of Randers Type
We consider the Finsler space $(\bar{M}^3, \bar{F})$ obtained by perturbing the Euclidean metric of $\mathbb{R}^3$ by a rotation. It is the open region of $\mathbb{R}^3$ bounded by a cylinder with a Randers metric. Using the Busemann-Hausdorff volume form, we obtain the differential equation that characterizes the helicoidal minimal surfaces in $\bar{M}^3$. We prove that the helicoid is a minimal surface in $\bar{M}^3$, only if the axis of the helicoid is the axis of the cylinder. Moreover, we prove that, in the Randers space $(\bar{M}^3, \bar{F})$, the only minimal surfaces in the Bonnet family, with fixed axis $O\bar{x}^3$, are the catenoids and the helicoids.

Keywords:minimal surfaces, helicoidal surfaces, Finsler space, Randers space
Categories:53A10, 53B40

3. CMB Online first

Parlier, Hugo
A short note on short pants
It is a theorem of Bers that any closed hyperbolic surface admits a pants decomposition consisting of curves of bounded length where the bound only depends on the topology of the surface. The question of the quantification of the optimal constants has been well studied and the best upper bounds to date are linear in genus, a theorem of Buser and Seppälä. The goal of this note is to give a short proof of a linear upper bound which slightly improve the best known bound.

Keywords:hyperbolic surfaces, geodesics, pants decompositions
Categories:30F10, 32G15, 53C22

4. CMB Online first

Jeong, Imsoon; Kim, Seonhui; Suh, Young Jin
Real Hypersurfaces in Complex Two-Plane Grassmannians with Reeb Parallel Structure Jacobi Operator
In this paper we give a characterization of a real hypersurface of Type~$(A)$ in complex two-plane Grassmannians ${ { {G_2({\mathbb C}^{m+2})} } }$, which means a tube over a totally geodesic $G_{2}(\mathbb C^{m+1})$ in ${G_2({\mathbb C}^{m+2})}$, by the Reeb parallel structure Jacobi operator ${\nabla}_{\xi}R_{\xi}=0$.

Keywords:real hypersurfaces, complex two-plane Grassmannians, Hopf hypersurface, Reeb parallel, structure Jacobi operator
Categories:53C40, 53C15

5. CMB 2013 (vol 57 pp. 401)

Perrone, Domenico
Curvature of $K$-contact Semi-Riemannian Manifolds
In this paper we characterize $K$-contact semi-Riemannian manifolds and Sasakian semi-Riemannian manifolds in terms of curvature. Moreover, we show that any conformally flat $K$-contact semi-Riemannian manifold is Sasakian and of constant sectional curvature $\kappa=\varepsilon$, where $\varepsilon =\pm 1$ denotes the causal character of the Reeb vector field. Finally, we give some results about the curvature of a $K$-contact Lorentzian manifold.

Keywords:contact semi-Riemannian structures, $K$-contact structures, conformally flat manifolds, Einstein Lorentzian-Sasaki manifolds
Categories:53C50, 53C25, 53B30

6. CMB 2012 (vol 57 pp. 209)

Zhao, Wei
Erratum to the Paper "A Lower Bound for the Length of Closed Geodesics on a Finsler Manifold"
We correct two clerical errors made in the paper "A Lower Bound for the Length of Closed Geodesics on a Finsler Manifold".

Keywords:Finsler manifold, closed geodesic, injective radius
Categories:53B40, 53C22

7. CMB 2012 (vol 57 pp. 194)

Zhao, Wei
A Lower Bound for the Length of Closed Geodesics on a Finsler Manifold
In this paper, we obtain a lower bound for the length of closed geodesics on an arbitrary closed Finsler manifold.

Keywords:Finsler manifold, closed geodesic, injective radius
Categories:53B40, 53C22

8. 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 metric
Categories:32V20, 53C56

9. 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 operator
Categories:53C15, 53C40

10. 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 quantity
Categories:53C60, 53B40

11. 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-curvature
Categories:53C60, 53B40

12. 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 flow
Categories:58C40, 53C44

13. 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 curvature
Categories:17B20, 22E46, 53C12

14. 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 submersion
Categories:53B20, 53C43

15. 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 theorems

16. 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, prequantization
Categories:53D, 22E

17. 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 flow
Categories:53C23, 28A35, 49Q20, 58A35

18. 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

19. 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, polystability
Categories:32L04, 53C07

20. CMB 2011 (vol 55 pp. 611)

Özgür, Cihan; Mihai, Adela
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 curvature
Categories:53C40, 53B05, 53B15

21. 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

22. 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 group
Categories:51M10, 32M15, 53C55, 53C35

23. 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 curvature
Categories:53B40, 53C60

24. 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 metric

25. 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 operator
Categories:53C40, 53C15
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