Symmetries of Contact Metric Manifolds

We study the Lie algebra of infinitesimal isometries on compact Sasakian and K-contact manifolds. On a Sasakian manifold which is not a space form or 3-Sasakian, every Killing vector field is an infinitesimal automorphism of the Sasakian structure. For a manifold with K-contact structure, we prove that there exists a Killing vector field of constant length which is not an infinitesimal automorphism of the structure if and only if the manifold is obtained from the Konishi bundle of a compact pseudo-Riemannian quaternion-Kähler manifold after changing the sign of the metric on a maximal negative distribution. We also prove that nonregular Sasakian manifolds are not homogeneous and construct examples with cohomogeneity one. Using these results we obtain in the last section the classification of all homogeneous Sasakian manifolds.     

Geodesic vector fields and Eikonal equation on a Riemannian manifold

In this paper, we study the impact of geodesic vector fields (vector fields whose trajectories are geodesics) on the geometry of a Riemannian manifold. Since, Killing vector fields of constant lengths on a Riemannian manifold are geodesic vector fields, leads to the question of finding sufficient conditions for a geodesic vector field to be Killing. In this paper, we show that a lower bound on the Ricci curvature of the Riemannian manifold in the direction of geodesic vector field gives a sufficient condition for the geodesic vector field to be Killing. Also, we use a geodesic vector field on a 3-dimensional complete simply connected Riemannian manifold to find sufficient conditions to be isometric to a 3-sphere. We find a characterization of an Einstein manifold using a Killing vector field. Finally, it has been observed that a major source of geodesic vector fields is provided by solutions of Eikonal equations on a Riemannian manifold and we obtain a characterization of the Euclidean space using an Eikonal equation.     

Stability numbers in K-contact manifolds

The aim of this paper is to study the stability of the characteristic vector field of a compact K-contact manifold with respect to the energy and volume functionals when we consider on the manifold a two-parameter variation of the metric. First of all, we multiply the metric in the direction of the characteristic vector field by a constant and then we change the metric by homotheties. We will study to what extent the results obtained in [V. Borrelli, Stability of the characteristic vector field of a Sasakian manifold, Soochow J. Math. 30 (2004) 283-292. Erratum on the article: Stability of the characteristic vector field of a Sasakian manifold, Soochow J. Math. 32 (2006) 179-180] for Sasakian manifolds are valid for a general K-contact manifold. Finally, as an example, we will study the stability of Hopf vector fields on Berger spheres when we consider homotheties of Berger metrics.     

Some Results on Contact Metric Manifolds

If the sectional curvatures of plane sections containing the characteristic vector field of a contact metric manifold M are non-vanishing, then we prove that a second order parallel tensor on M is a constant multiple of the associated metric tensor. Next, we prove for a contact metric manifold of dimension greater than 3 and whose Ricci operator commutes with the fundamental collineation that, if its Weyl conformal tensor is harmonic, then it is Einstein. We also prove that, if the Lie derivative of the fundamental collineation along the characteristic vector field on a contact metric 3-manifold M satisfies a cyclic condition, then M is either Sasakian or locally isometric to certain canonical Lie-groups with a left invariant metric. Next, we prove that if a three-dimensional Sasakian manifold admits a non-Killing projective vector field, it is of constant curvature 1. Finally, we prove that a conformally recurrent Sasakian manifold is locally isometric to a unit sphere.     

向量场的Nielsen数

对于紧致流形M上的任意一个向量场X，定义了一个由向量场X确定的自映射fX：M→M，使得向量场X的奇异点均为fX的不动点．证明了向量场的Nielsen数是不依赖于向量场选取的量．     

Holomorphic vector fields with totally degenerate zeroes

A zero set of a holomorphic vector field is totally degenerate, if the endomorphism of the conormal sheaf induced by the vector field is identically zero. By studying a class of foliations generalizing foliations of C^*-actions, we show that if a projective manifold admits a holomorphic vector field with a smooth totally degenerate zero component,then the manifold is stably birational to that component of the zero set.When the vector field has an isolated totally degenerate zero, we prove that the manifold is rational. This is a special case of Carrell's conjecture.     

Almost contact metric manifolds whose Reeb vector field is a harmonic section

We investigate almost contact metric manifolds whose Reeb vector field is a harmonic unit vector field, equivalently a harmonic section. We first consider an arbitrary Riemannian manifold and characterize the harmonicity of a unit vector field ??, when ??? is symmetric, in terms of Ricci curvature. Then, we show that for the class of locally conformal almost cosymplectic manifolds whose Reeb vector field ?? is geodesic, ?? is a harmonic section if and only if it is an eigenvector of the Ricci operator. Moreover, we build a large class of locally conformal almost cosymplectic manifolds whose Reeb vector field is a harmonic section. Finally, we exhibit several classes of almost contact metric manifolds where the associated almost contact metric structures ?? are harmonic sections, in the sense of Vergara-Diaz and Wood?[25], and in some cases they are also harmonic maps.     

On semi-parallel lightlike hypersurfaces of indefinite Kenmotsu manifolds

We study semi-parallel lightlike hypersurfaces of an indefinite Kenmotsu manifold, tangent to the structure vector field. Some Theorems on parallel and semi-parallel vector field, geodesibility of lightlike hypersurfaces are obtained. The geometrical configuration of such lightlike hypersurfaces is established. We prove that, in totally contact umbilical lightlike hypersurfaces of an indefinite Kenmotsu manifold which has constant ${\overline{\phi}}$ -holomorphic sectional curvature c, tangent to the structure vector field and such that its distribution is parallel, the parallelism and semi-parallelism notions are equivalent.     

Automorphism invariant Cartan subgroups and power maps of disconnected groups

We study the geometry and the periodic geodesics of a compact Lorentzian manifold that has a Killing vector field which is timelike somewhere. Using a compactness argument for subgroups of the isometry group, we prove the existence of one timelike non self-intersecting periodic geodesic. If the Killing vector field is nowhere vanishing, then there are at least two distinct periodic geodesics; as a special case, compact stationary manifolds have at least two periodic timelike geodesics. We also discuss some properties of the topology of such manifolds. In particular, we show that a compact manifold M admits a Lorentzian metric with a nowhere vanishing Killing vector field which is timelike somewhere if and only if M admits a smooth circle action without fixed points.     

Sasakian切触度量(κ,μ)-空间中子流形的分类

特征矢量场属于某（κ，μ）－幂零分布的切触度量流形称为切触度量（κ，μ）空间，本文中我们证明了当κ^2 μ^2≠0时，一个非Sasakian切触度量（κ，μ）－空间中的任何子流形要么是不变的全测地子流形。要么为反不变子流形。     

Conformal vector fields on Kaehler manifolds

It is known that a conformal vector field on a compact Kaehler manifold is a Killing vector field. In this paper, we are interested in finding conditions under which a conformal vector field on a non-compact Kaehler manifold is Killing. First we prove that a harmonic analytic conformal vector field on a 2n-dimensional Kaehler manifold (n ≠ 2) of constant nonzero scalar curvature is Killing. It is also shown that on a 2n-dimensional Kaehler Einstein manifold (n > 1) an analytic conformal vector field is either Killing or else the Kaehler manifold is Ricci flat. In particular, it follows that on non-flat Kaehler Einstein manifolds of dimension greater than two, analytic conformal vector fields are Killing.     

On almost pseudo-conformally symmetric Ricci-recurrent manifolds with applications to relativity

The object of the present paper is to study almost pseudo-conformally symmetric Ricci-recurrent manifolds. The existence of almost pseudo-conformally symmetric Ricci-recurrent manifolds has been proved by an explicit example. Some geometric properties have been studied. Among others we prove that in such a manifold the vector field ? corresponding to the 1-form of recurrence is irrotational and the integral curves of the vector field ? are geodesic. We also study some global properties of such a manifold. Finally, we study almost pseudo-conformally symmetric Ricci-recurrent spacetime. We obtain the Segre’ characteristic of such a spacetime.     

Killing vector fields of horizontal Liouville type

On a slit tangent bundle endowed with a Riemannian metric of Sasaki–Finsler type, we introduce two vector fields of horizontal Liouville type and prove that these vector fields are Killing if and only if the base Finsler manifold is of positive constant curvature. In the special case of one of them, we show that if it is Killing vector field then the base manifold is Einstein–Finsler manifold.     

Halpern- and Mann-Type Algorithms for Fixed Points and Inclusion Problems on Hadamard Manifolds

In this article, we consider an inclusion problem which is defined by means of a sum of a single-valued vector field and a set-valued vector field defined on a Hadamard manifold. We propose Halpern-type and Mann-type algorithms for finding a common point of the set of fixed points of a nonexpansive mapping and the set of solutions of the inclusion problem defined on a Hadamard manifold. Some particular cases of our problem and algorithm are also discussed. We study the convergence of the proposed algorithm to a common point of the set of fixed points of a nonexpansive mapping and the set of solutions of the inclusion problem defined on a Hadamard manifold. As applications of our results and algorithms, we derive the solution methods and their convergence results for the optimization problems, variational inequality problems and equilibrium problems in the setting of Hadamard manifolds.     

Computing (Un)stable Manifolds with Validated Error Bounds: Non-resonant and Resonant Spectra

We develop techniques for computing the (un)stable manifold at a hyperbolic equilibrium of an analytic vector field. Our approach is based on the so-called parametrization method for invariant manifolds. A feature of this approach is that it leads to a posteriori analysis of truncation errors which, when combined with careful management of round off errors, yields a mathematically rigorous enclosure of the manifold. The main novelty of the present work is that, by conjugating the dynamics on the manifold to a polynomial rather than a linear vector field, the computer-assisted analysis is successful even in the case when the eigenvalues fail to satisfy non-resonance conditions. This generically occurs in parametrized families of vector fields. As an example, we use the method as a crucial ingredient in a computational existence proof of a connecting orbit in an amplitude equation related to a pattern formation model that features eigenvalue resonances.     

Clairaut Anti-invariant Submersions from Sasakian and Kenmotsu Manifolds

We investigate new Clairaut conditions for anti-invariant submersions from Sasakian and Kenmotsu manifolds onto Riemannian manifolds. We prove that there do not exist Clairaut anti-invariant submersion admitting vertical Reeb vector field in case of the total manifold is Sasakian. Several illustrative examples are also included.     

On conformal Killing symmetric tensor fields on Riemannian manifolds

A vector field on Riemannian manifold is called conformal Killing if it generates oneparameter group of conformal transformation. The class of conformal Killing symmetric tensor fields of an arbitrary rank is a natural generalization of the class of conformal Killing vector fields, and appears in different geometric and physical problems. In this paper, we prove that a trace-free conformal Killing tensor field is identically zero if it vanishes on some hypersurface. This statement is a basis of the theorem on decomposition of a symmetric tensor field on a compact manifold with boundary to a sum of three fields of special types. We also establish triviality of the space of trace-free conformal Killing tensor fields on some closed manifolds.     

Conformal fields and the stability of leaves with constant higher order mean curvature

In this paper, we study hypersurfaces with constant rth mean curvature Sr. We investigate the stability of such hypersurfaces in the case when they are leaves of a codimension one foliation. We also generalize recent results by Barros and Sousa, concerning conformal fields, to an arbitrary manifold. Using this we show that normal component of a Killing field is an rth Jacobi field of a hypersurface with Sr+1 constant. Finally, we study relations between rth Jacobi fields and vector fields preserving a foliation.     

On the Curvature Properties of Manifolds with Deformed <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript> Structures

In this paper, we study the curvature properties of a manifold with structure group G₂ whose fundamental 3-form is deformed by a Killing vector field of unit length. We obtain some results concerning conditions under which this manifold is flat, Einstein, or isometric to the unit sphere.     

Contact metric manifolds whose characteristic vector field is a harmonic vector field

In this paper, contact metric manifolds whose characteristic vector field ξ is a harmonic vector field are called H-contact manifolds. We show that a (2n+1)-dimensional contact metric manifold is an H-contact manifold if and only if ξ is an eigenvector of the Ricci operator (J.C. González-Dávila and L. Vanhecke [J. Geom. 72 (2001) 65–76] proved this result for n=1). Consequently, the class of H-contact manifolds is very large. Moreover, we give some application about the topology of a compact H-contact manifold.