Expand all Collapse all | Results 1 - 25 of 82 |
1. CMB Online first
On the maximum curvature of closed curves in negatively curved manifolds Motivated by Almgren's work on the isoperimetric inequality,
we prove a sharp inequality relating the length and maximum curvature
of a closed curve in a complete, simply connected manifold of
sectional curvature at most $-1$. Moreover, if equality holds,
then the norm of the geodesic curvature is constant and the torsion
vanishes. The proof involves an application of the maximum principle
to a function defined on pairs of points.
Keywords:manifold, curvature Category:53C20 |
2. CMB Online first
Sur les algÃ¨bres de Lie associÃ©es Ã une connexion Let $\Gamma$ be a connection on a smooth manifold
$M$, in this paper we give some properties of $\Gamma$ by studying
the corresponding Lie algebras. In particular, we compute the
first Chevalley-Eilenberg cohomology space of the horizontal
vector fields Lie algebra on the tangent bundle of $M$, whose
the corresponding Lie derivative of $\Gamma$ is null, and of
the horizontal nullity curvature space.
Keywords:algÃ¨bre de Lie, connexion, cohomologie de Chevalley-Eilenberg, champs dont la dÃ©rivÃ©e de Lie correspondante Ã une connexion est nulle, espace de nullitÃ© de la courbure Categories:17B66, 53B15 |
3. CMB Online first
The Diffeomorphism Type of Canonical Integrations Of Poisson tensors on Surfaces A surface $\Sigma$ endowed with a Poisson tensor
$\pi$ is known to admit
canonical integration, $\mathcal{G}(\pi)$,
which is a 4-dimensional manifold with a (symplectic) Lie groupoid
structure.
In this short note we show that if $\pi$ is not an area
form on the 2-sphere, then $\mathcal{G}(\pi)$ is diffeomorphic
to the cotangent bundle $T^*\Sigma$. This extends
results by the author and by Bonechi, Ciccoli, Staffolani, and Tarlini.
Keywords:Poisson tensor, Lie groupoid, cotangent bundle Categories:58H05, 55R10, 53D17 |
4. CMB Online first
Cohomogeneity one Randers metrics An action of a Lie group $G$ on a smooth manifold $M$ is called
cohomogeneity one if the orbit space $M/G$ is of dimension $1$.
A Finsler metric $F$ on $M$ is called invariant if $F$ is
invariant under the action of $G$. In this paper,
we study invariant
Randers metrics on cohomogeneity one manifolds. We first give a
sufficient and necessary condition for the existence of invariant
Randers metrics on cohomogeneity one manifolds. Then we obtain
some results on invariant Killing vector fields on the
cohomogeneity one manifolds and use that to deduce some
sufficient and necessary condition for a cohomogeneity one
Randers metric to be Einstein.
Keywords:cohomogeneity one actions, normal geodesics, invariant vector fields, Randers metrics Categories:53C30, 53C60 |
5. CMB Online first
Ricci Curvature Tensor and Non-Riemannian Quantities There are several notions of Ricci curvature tensor
in Finsler geometry and spray geometry. One of them is defined by the
Hessian of the well-known Ricci curvature.
In this paper we will introduce a new notion of Ricci curvature
tensor and discuss its relationship with the Ricci curvature and some
non-Riemannian quantities. By this Ricci curvature tensor, we shall
have a better understanding on these non-Riemannian quantities.
Keywords:Finsler metrics, sprays, Ricci curvature, non-Riemanian quantity Categories:53B40, 53C60 |
6. CMB Online first
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 |
7. CMB 2014 (vol 58 pp. 158)
Corrigendum to "Chen Inequalities for Submanifolds of Real Space Forms with a Semi-symmetric Non-metric Connection" |
Corrigendum to "Chen Inequalities for Submanifolds of Real Space Forms with a Semi-symmetric Non-metric Connection" We fix the coefficients in the inequality (4.1) in the Theorem 4.1(i) from
A. Mihai and C. ÃzgÃ¼r, "Chen inequalities for
submanifolds of real space forms with a semi-symmetric non-metric
connection" Canad. Math. Bull. 55 (2012), no. 3, 611-622.
Keywords:real space form, semi-symmetric non-metric connection, Ricci curvature Categories:53C40, 53B05, 53B15 |
8. CMB 2014 (vol 57 pp. 765)
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 |
9. CMB 2013 (vol 57 pp. 870)
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 SeppÃ¤lÃ¤. 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 |
10. CMB 2013 (vol 57 pp. 821)
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 |
11. CMB 2013 (vol 57 pp. 401)
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 |
12. CMB 2012 (vol 57 pp. 209)
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 |
13. CMB 2012 (vol 57 pp. 194)
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 |
14. CMB 2012 (vol 57 pp. 12)
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 |
15. CMB 2011 (vol 56 pp. 306)
Real Hypersurfaces in Complex Projective Space Whose Structure Jacobi Operator is Lie $\mathbb{D}$-parallel |
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 |
16. CMB 2011 (vol 56 pp. 184)
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 |
17. CMB 2011 (vol 56 pp. 615)
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 |
18. CMB 2011 (vol 56 pp. 127)
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 |
19. CMB 2011 (vol 55 pp. 870)
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 |
20. CMB 2011 (vol 56 pp. 173)
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 |
21. CMB 2011 (vol 55 pp. 632)
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 Category:53C20 |
22. CMB 2011 (vol 56 pp. 116)
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 |
23. CMB 2011 (vol 55 pp. 723)
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 |
24. CMB 2011 (vol 55 pp. 663)
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 |
25. CMB 2011 (vol 56 pp. 44)
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 |