Transformations of locally conformally Kähler manifolds

We consider several transformation groups of a locally conformally Kähler manifold and discuss their inter-relations. Among other results, we prove that all conformal vector fields on a compact Vaisman manifold which is neither locally conformally hyperkähler nor a diagonal Hopf manifold are Killing, holomorphic and that all affine vector fields with respect to the minimal Weyl connection of a locally conformally Kähler manifold which is neither Weyl-reducible nor locally conformally hyperkähler are holomorphic and conformal.     

Almost Hermitian manifolds admitting holomorphically planar conformal vector fields

We classify and characterize an almost Hermitian manifold M admitting a holomorphically planar conformal vector (HPCV) field (a generalization of a closed conformal vector field) V . We show that if V is nowhere vanishing and strictly non-geodesic, then it is homothetic and almost analytic. If, in addition,M satisfies Gray’s first condition, then M is Kaehler. For a semi-Kaehler manifold M admitting an HPCV field V we show that either V is closed, or M becomes almost Kaehler and V is homothetic and almost analytic. Part of this work was done by the second author while he was visiting Sri Sathya Sai Institute Of Higher Learning, Prasanthinilayam, India.     

Conformal holonomy of C-spaces, Ricci-flat, and Lorentzian manifolds

The main result of this paper is that a Lorentzian manifold is locally conformally equivalent to a manifold with recurrent lightlike vector field and totally isotropic Ricci tensor if and only if its conformal tractor holonomy admits a 2-dimensional totally isotropic invariant subspace. Furthermore, for semi-Riemannian manifolds of arbitrary signature we prove that the conformal holonomy algebra of a C-space is a Berger algebra. For Ricci-flat spaces we show how the conformal holonomy can be obtained by the holonomy of the ambient metric and get results for Riemannian manifolds and plane waves.     

On conformal vector fields on Randers manifolds

In this paper,we study conformal vector fields on a Randers manifold with certain curvature properties.In particular,we completely determine conformal vector fields on a Randers manifold of weakly isotropic flag curvature.     

Conformal vector fields on some Finsler manifolds

We study conformal vector fields on a Finsler manifold whose metric is defined by a Riemannian metric, a 1-form and its norm. We find PDEs characterizing conformal vector fields. Then we obtain the explicit expressions of conformal vector fields for certain spherically symmetric metrics on R~n.     

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.     

On causality and closed geodesics of compact Lorentzian manifolds and static spacetimes

Some results related to the causality of compact Lorentzian manifolds are proven: (1) any compact Lorentzian manifold which admits a timelike conformal vector field is totally vicious, and (2) a compact Lorentzian manifold covered regularly by a globally hyperbolic spacetime admits a timelike closed geodesic, if some natural topological assumptions (fulfilled, for example, if one of the conjugacy classes of deck transformations containing a closed timelike curve is finite) hold. As a consequence, any compact Lorentzian manifold conformal to a static spacetime is geodesically connected by causal geodesics, and admits a timelike closed geodesic.     

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.     

Projective geometry on the bundle of volume forms

The bundle of volume forms on a manifold is examined in terms of the affine connection defined by T. Y. Thomas. The choice of a particular affine connection in the projective class corresponds to the choice of an horizontal distribution on this bundle. The geometric properties of the horizontal distributions are studied. Special lifts of vector fields and covariant tensor fields are examined as well as lifts of metric connections.     

Projective vector fields on Finsler manifolds

In this paper, we give the equation that characterizes projective vector fields on a Finsler manifold by the local coordinate. Moreover, we obtain a feature of the projective fields on the compact Finsler manifold with non-positive flag curvature and the non-existence of projective vector fields on the compact Finsler manifold with negative flag curvature. Furthermore, we deduce some expectable, but non-trivial relationships between geometric vector fields such as projective, affine, conformal, homothetic and Killing vector fields on a Finsler manifold.     

On geometric vector fields of Minkowski spaces and their applications

As it is well-known, a Minkowski space is a finite dimensional real vector space equipped with a Minkowski functional F. By the help of its second order partial derivatives we can introduce a Riemannian metric on the vector space and the indicatrix hypersurface S:=F⁻¹(1) can be investigated as a Riemannian submanifold in the usual sense.Our aim is to study affine vector fields on the vector space which are, at the same time, affine with respect to the Funk metric associated with the indicatrix hypersurface. We give an upper bound for the dimension of their (real) Lie algebra and it is proved that equality holds if and only if the Minkowski space is Euclidean. Criteria of the existence is also given in lower dimensional cases. Note that in case of a Euclidean vector space the Funk metric reduces to the standard Cayley-Klein metric perturbed with a nonzero 1-form.As an application of our results we present the general solution of Matsumoto's problem on conformal equivalent Berwald and locally Minkowski manifolds. The reasoning is based on the theory of harmonic vector fields on the tangent spaces as Riemannian manifolds or, in an equivalent way, as Minkowski spaces. Our main result states that the conformal equivalence between two Berwald manifolds must be trivial unless the manifolds are Riemannian.     

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.     

Self-duality operators on odd dimensional manifolds

In this paper we construct a new elliptic operator associated to any nowhere zero vector field on an odd-dimensional manifold and study its index theory. It turns out this operator has several geometric applications to conformal vector fields, self-dual vector fields, locally free -actions and transversal hypersurfaces of these vector fields in an odd-dimensional manifold. In particular, we reveal a non-stable phenomena about the existence of conformal vector fields and self-dual vector fields in odd dimensions above 3. This is in sharp contrast to the stable phenomena about the existence of nowhere zero vector fields in odd dimensions. Besides these applications, the index formula of this new operator also gives the formulas for the dimensions of self-duality cohomology groups and for the virtual dimensions of the moduli spaces of anti-self-dual connections on 5-cobordisms, which are introduced in author's previous papers.

     

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.     

Rigidity of ‐quasi Einstein manifolds

This paper deals with the study on ‐quasi Einstein manifolds. First, we give some characterizations of an ‐quasi Einstein manifold admitting closed conformal or parallel vector field. Then, we obtain some rigidity conditions for this class of manifolds. We prove that an ‐quasi Einstein manifold with a closed conformal vector field has a warped product structure of the form , where I is a real interval, is an ‐dimensional Riemannian manifold and q is a smooth function on I . Finally, a non‐trivial example of an ‐quasi Einstein manifold verifying our results in terms of the potential function is presented.     

Ambient connections realising conformal Tractor holonomy

For a conformal manifold we introduce the notion of an ambient connection, an affine connection on an ambient manifold of the conformal manifold, possibly with torsion, and with conditions relating it to the conformal structure. The purpose of this construction is to realise the normal conformal Tractor holonomy as affine holonomy of such a connection. We give an example of an ambient connection for which this is the case, and which is torsion free if we start the construction with a C-space, and in addition Ricci-flat if we start with an Einstein manifold. Thus, for a C-space this example leads to an ambient metric in the weaker sense of Čap and Gover, and for an Einstein space to a Ricci-flat ambient metric in the sense of Fefferman and Graham. Current address for first author: Erwin Schr?dinger International Institute for Mathematical Physics (ESI), Boltzmanngasse 9, 1090 Vienna, Austria Current address for second author: Department of Mathematics, University of Hamburg, Bundesstra?e 55, 20146 Hamburg, Germany     

Global hyperbolicity and Palais-Smale condition for action functionals in stationary spacetimes

In order to apply variational methods to the action functional for geodesics of a stationary spacetime, some hypotheses, useful to obtain classical Palais-Smale condition, are commonly used: pseudo-coercivity, bounds on certain coefficients of the metric, etc. We prove that these technical assumptions admit a natural interpretation for the conformal structure (causality) of the manifold. As a consequence, any stationary spacetime with a complete timelike Killing vector field and a complete Cauchy hypersurface (thus, globally hyperbolic), is proved to be geodesically connected.     

Indefinite locally conformal Kähler manifolds

We study the basic properties of an indefinite locally conformal Kähler (l.c.K.) manifold. Any indefinite l.c.K. manifold M with a parallel Lee form ω is shown to possess two canonical foliations F and Fc, the first of which is given by the Pfaff equation ω=0 and the second is spanned by the Lee and the anti-Lee vectors of M. We build an indefinite l.c.K. metric on the noncompact complex manifold Ω₊=(Λ₊?Λ₀)/Gλ (similar to the Boothby metric on a complex Hopf manifold) and prove a CR extension result for CR functions on the leafs of F when M=Ω₊ (where is −²|z₁|−?−²|zs|+²|zs+1|+?+²|zn|>0). We study the geometry of the second fundamental form of the leaves of F and Fc. In the degenerate cases (corresponding to a lightlike Lee vector) we use the technique of screen distributions and (lightlike) transversal bundles developed by A. Bejancu et al. [K.L. Duggal, A. Bejancu, Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, vol. 364, Kluwer Academic, Dordrecht, 1996].     

Affine conformal vector fields in semi-Riemannian manifolds

This paper is devoted to a systematic presentation of the essential results of research on affine conformal vector fields (ACV) and to exhibit the state of art as it now stands. Of particular interest is the new information on the existence of ACVs in compact orientable semi-Riemannian manifolds, their link with first integrals of the geodesics and the separability structures.