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.     

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.     

Contact totally umbilical submanifolds of a sasakian space form

Summary We define a notion of contact totally umbilical submanifolds of Sasakian space forms corresponds to those of totally umbilical submanifolds of complex space forms. We study a contact totally umbilical submanifold M of a Sasakian space form (c ≠ −3) and prove that M is an invariant submanifold or an anti-invariant submanifold. Furthermore we study a submanifold M with parallel second fundamental form of a Sasakian space form (c ≠ 1) and prove that M is invariant or anti-invariant. Entrata in Redazione il 7 settembre 1976.     

On Killing tensors in three-dimensional Euclidean space

Theoretical and Mathematical Physics - We discuss the properties of second-order Killing tensors in three-dimensional Euclidean space that guarantee the existence of a third integral of motion...     

On the nullity of conformal Killing graphs in foliated Riemannian spaces

We deal with entire conformal Killing graphs, that is, graphs constructed through the flow generated by a complete conformal Killing vector field V on a Riemannian space \({\overline{M}}\) , and which are defined over an integral leaf of the foliation \({V^\bot {\rm of} \overline{M}}\) orthogonal to V. Under a suitable restriction on the norm of the gradient of the function z which determines such a graph Σ(z), we establish sufficient conditions to ensure that Σ(z) is totally geodesic. Afterwards, when the ambient space \({\overline{M}}\) has constant sectional curvature, we obtain lower estimates for the index of minimum relative nullity of Σ(z).     

On gradient Ricci solitons conformal to a pseudo-Euclidean space

We consider gradient Ricci solitons, conformal to an n-dimensional pseudo-Euclidean space, which are invariant under the action of an (n ? 1)-dimensional translation group. We provide all such solutions in the case of steady gradient Ricci 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.     

On the Riemannian curvature tensor of a real hypersurface in a complex projective space

We consider real hypersurfaces M in complex projective space equipped with both the Levi–Civita and generalized Tanaka–Webster connections and classify them when the covariant derivatives associated with both connections, either in the direction of the structure vector field or any direction of the maximal holomorphic distribution, coincide when applying to the Riemannian curvature tensor of the real hypersurface.     

Generalized Korn's inequality and conformal Killing vectors

Korn's inequality plays an important role in linear elasticity theory. This inequality bounds the norm of the derivatives of the displacement vector by the norm of the linearized strain tensor. The kernel of the linearized strain tensor are the infinitesimal rigid-body translations and rotations (Killing vectors). We generalize this inequality by replacing the linearized strain tensor by its trace free part. That is, we obtain a stronger inequality in which the kernel of the relevant operator are the conformal Killing vectors. The new inequality has applications in General Relativity.     

On the ricci tensor of a real hypersurface of quaternionic hyperbolic space

Summary We prove the non-existence of Einstein real hypersurfaces of quaternionic hyperbolic space. This article was processed by the author using the IAT_EX style filecljour1 from Springer-Verlag.     

Killing vector fields and the curvature tensor of a Riemannian manifold

We find a convenient expression for the value of the covariant curvature 4-tensor of an arbitrary Riemannian manifold on a quadruple of its Killing vector fields. With its use, we in particular obtain a simple deduction of the well-known formula to calculate the sectional curvature of a homogeneous Riemannian space.     

The analogue of Weyl's conformal curvature tensor in a Michal functional geometry

The purpose of this note is to obtain a set of functionals conformally invariant in a Michal functional geometry. The author is indebted to prof.A. D. Michal for calling his attention to this problem and for helpful suggestions and criticisms in the preparation of this paper.     

The Killing vector and the generalised Killing equation in Finsler space

Rendiconti del Circolo Matematico di Palermo Series 2 - Recently we have studied the infinitesimal deformation in Finsler space [8] (1). Here we have obtained the necessary and sufficient condition...     

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.     

Killing helices on a symmetric space of rank one

In this paper, we study helices which are orbits of one parameter families of isometries on a symmetric space of rank one. We introduce the notion of structure torsion fields for helices, show the necessary and sufficient condition that they are generated by some Killing vector fields, and study their moduli space. The author is partially supported by Grant-in-Aid for Scientific Research (C)(No. 14540075), Ministry of Education, Science, Sports, Culture and Technology.