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.     

The conformal Killing equation on forms—prolongations and applications

We construct a conformally invariant vector bundle connection such that its equation of parallel transport is a first order system that gives a prolongation of the conformal Killing equation on differential forms. Parallel sections of this connection are related bijectively to solutions of the conformal Killing equation. We construct other conformally invariant connections, also giving prolongations of the conformal Killing equation, that bijectively relate solutions of the conformal Killing equation on k-forms to a twisting of the conformal Killing equation on (k−?)-forms for various integers ?. These tools are used to develop a helicity raising and lowering construction in the general setting and on conformally Einstein manifolds.     

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.     

Some conformal and projective scalar invariants of Riemannian manifolds

It is proved that, on any closed oriented Riemannian n-manifold, the vector spaces of conformal Killing, Killing, and closed conformal Killing r-forms, where 1 ≤ r ≤ n ? 1, have finite dimensions t _r , k _r , and p _r , respectively. The numbers t _r are conformal scalar invariants of the manifold, and the numbers k _r and p _r are projective scalar invariants; they are dual in the sense that t _r = t _n?r and k _r = p _n?r . Moreover, an explicit expression for a conformal Killing r-form on a conformally flat Riemannian n-manifold is given.     

On Conformal Infinity and Compactifications of the Minkowski Space

Using the standard Cayley transform and elementary tools it is reiterated that the conformal compactification of the Minkowski space involves not only the “cone at infinity” but also the 2-sphere that is at the base of this cone. We represent this 2-sphere by two additionally marked points on the Penrose diagram for the compactified Minkowski space. Lacks and omissions in the existing literature are described, Penrose diagrams are derived for both, simple compactification and its double covering space, which is discussed in some detail using both the U(2) approach and the exterior and Clifford algebra methods. Using the Hodge *{\star} operator twistors (i.e. vectors of the pseudo-Hermitian space H _2,2) are realized as spinors (i.e., vectors of a faithful irreducible representation of the even Clifford algebra) for the conformal group SO(4, 2)/Z ₂. Killing vector fields corresponding to the left action of U(2) on itself are explicitly calculated. Isotropic cones and corresponding projective quadrics in H _p,q are also discussed. Applications to flat conformal structures, including the normal Cartan connection and conformal development has been discussed in some detail.     

The Vector Space of Conformal Killing Forms on a Riemannian Manifold

T. Kashiwada and S. Tachibana defined conformal Killing p-forms on a Riemannian manifold of dimension p \geqslant 1$$ " align="middle" border="0"> and generalized some results on conformal Killing vector fields to the case of such forms. In this paper, conformal Killing p-forms are defined with the help of natural differental operators on Riemannian manifolds and representations of orthogonal groups. The geometry of the vector space of conformal Killing p-forms and of its two subspaces of coclosed conformal Killing p-forms and of closed conformal Killing p-forms are considered. Some local and global results due to Tachibana and Kashiwada about conformal Killing and Killing p-forms are generalized. An application to Hermitian geometry is given. Bibliography: 30 titles.     

Local dynamics of conformal vector fields

We study pseudo-Riemannian conformal vector fields in the neighborhood of a singularity. For Riemannian manifolds, we prove that if a conformal vector field vanishing at a point x ₀ is not Killing for a metric in the conformal class, then a neighborhood of the singularity x ₀ is conformally flat.     

A note on Kazdan-Warner-type identities

In this article we deal with f-conformal Killing vector fields, a generalization of conformal Killing vector fields involving a function f and two real parameters. Under certain conditions on f and the parameters, some non-existence results for such vector fields are proven. Moreover we derive a generalization of the Kazdan-Warner and Bourguignon-Ezin identities to the case of f-conformal Killing vector fields. Based on these, finally we present an application to the f-conformal solitons, a generalization of the Yamabe solitons.     

Korn's inequality and Donati's theorem for the conformal Killing operator on pseudo-Euclidean space

We prove the Korn's inequality for the conformal Killing operator on pseudo-Euclidean space Rp,q, and an existence theorem for solutions to the non-homogeneous conformal Killing equation, which is a pseudo-Euclidean conformal generalization of Donati's theorem for Euclidean Killing operator.     

Laplacian Operators and Q-curvature on Conformally Einstein Manifolds

A new definition of canonical conformal differential operators P _k (k = 1,2,...), with leading term a kth power of the Laplacian, is given for conformally Einstein manifolds of any signature. These act between density bundles and, more generally, between weighted tractor bundles of any rank. By construction these factor into a power of a fundamental Laplacian associated to Einstein metrics. There are natural conformal Laplacian operators on density bundles due to Graham–Jenne–Mason–Sparling (GJMS). It is shown that on conformally Einstein manifolds these agree with the P _k operators and hence on Einstein manifolds the GJMS operators factor into a product of second-order Laplacian type operators. In even dimension n the GJMS operators are defined only for 1 ≤ k ≤ n/2 and so, on conformally Einstein manifolds, the P _k give an extension of this family of operators to operators of all even orders. For n even and k > n/2 the operators P _k are each given by a natural formula in terms of an Einstein metric but they are not natural conformally invariant operators in the usual sense. They are shown to be nevertheless canonical objects on conformally Einstein structures. There are generalisations of these results to operators between weighted tractor bundles. It is shown that on Einstein manifolds the Branson Q-curvature is constant and an explicit formula for the constant is given in terms of the scalar curvature. As part of development, conformally invariant tractor equations equivalent to the conformal Killing equation are presented.     

λ-Automorphisms of a Riemannian foliation

We study geometric properties of -automorphisms of a Riemannian foliationF which is not harmonic. This notion was first introduced in [KTT] for the case whereF is harmonic. Transversal Killing, affine, conformal, projective fields are all examples of -automorphisms. We derive several general identities for a -automorphism. In particular, we extend the results on the transversal conformal and Killing fields obtained in [PrY], [NY1,2]. Furthermore, we analyse the geometric meaning of the condition appearing in our results.The present studies were supported (in part) by the Basic Science Research Institute Program, Ministry of Education, 1994, Project No. BSRI-94-1404     

Two-jets of conformal fields along their zero sets

The connected components of the zero set of any conformal vector field v, in a pseudo-Riemannian manifold (M, g) of arbitrary signature, are of two types, which may be called ‘essential’ and ‘nonessential’. The former consist of points at which v is essential, that is, cannot be turned into a Killing field by a local conformal change of the metric. In a component of the latter type, points at which v is nonessential form a relatively-open dense subset that is at the same time a totally umbilical submanifold of (M, g). An essential component is always a null totally geodesic submanifold of (M, g), and so is the set of those points in a nonessential component at which v is essential (unless this set, consisting precisely of all the singular points of the component, is empty). Both kinds of null totally geodesic submanifolds arising here carry a 1-form, defined up to multiplications by functions without zeros, which satisfies a projective version of the Killing equation. The conformal-equivalence type of the 2-jet of v is locally constant along the nonessential submanifold of a nonessential component, and along an essential component on which the distinguished 1-form is nonzero. The characteristic polynomial of the 1-jet of v is always locally constant along the zero set.     

A x -operator on complete riemannian manifolds

In this paper we give a generalisation of Kostant’s Theorem about theA _x -operator associated to a Killing vector fieldX on a compact Riemannian manifold. Kostant proved (see [6], [5] or [7]) that in a compact Riemannian manifold, the (1, 1) skew-symmetric operatorA _x =L _x −≡ _x associated to a Killing vector fieldX lies in the holonomy algebra at each point. We prove that in a complete non-compact Riemannian manifold (M, g) theA _x -operator associated to a Killing vector field, with finite global norm, lies in the holonomy algebra at each point. Finally we give examples of Killing vector fields with infinite global norms on non-flat manifolds such thatA _x does not lie in the holonomy algebra at any point.     

Some applications of globally framed structures to relativity

Summary It is shown that a class of 4-dimensional Lorentzian framed space-times (defined by a linear operator satisfying certain algebraic and geometric conditions) admitting a nonsingular simple electromagnetic field possesses a two parameter abelian group of affine conformal motions. Based on this, we have studied the problem of finding various types of inheriting electromagnetic field plus perfect fluid solutions. A sub-class of such spaces exists whose geometry is conformal to a physical space-time having an invertible two parameter abelian isometry group. Thus, our work is related to the Carter's theorems on Killing Horizons.     

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.     

Generalized killing tensors of arbitrary rank and order

We define Killing tensors and conformal Killing tensors of arbitrary rank and order which generalize in a natural way the notion of a Killing vector. We explicitly derive the corresponding tensors for a flat de Sitter space of dimension p+q=m,m 4, which permits us to calculate complete sets of symmetry operators of arbitrary order n for a scalar wave equation with m independent parameters.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 6, pp. 786–795, June, 1991.     

On a conformal Killingp-form in a compact sasakian space

Summary Let M be a compact Sasakian space admitting a conformal Killing p-form u. Then, we show that the associated form ϑ of a conformal Killing form u is a special Killing form with constant 1. Moreover we prove the decomposition theorem of u and seek the condition for M to be a unit sphere. Entrata in Redazione il 29 agosto 1971. Delicated to ProfessorT. Adati on his 60th Birthday.     

Some properties of locally conformal symplectic structures

We show that the d_w d_{\omega} -cohomology is isomorphic to a conformally invariant usual de Rham cohomology of an appropriate cover. We also prove a Moser theorem for locally conformal symplectic (lcs) forms. We point out a connection between lcs geometry and contact geometry. Finally, we show the connections between first kind, second kind, essential, inessential, local, and global conformal symplectic structures through several invariants.     

On a horizontal conformal Killing tensor of degreep in a sasakian space

Summary We deal with a horizontal conformal Killing tensor of degree p in a Sasakian space. After some preparations we prove that a horizontal conformal Killing tensor of odd degree is necessarily Killing. Moreover, we consider horizontal conformal Killing tensor of even degree. The form of the associated tensor is determined completely and a decomposition theorem is proved. Then we give the examples of a conformal Killing tensor of even degree and a special Killing tensor of odd degree with constant l. Entrata in Redazione il 17 luglio 1971.