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.     

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.     

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.     

A Bernstein-type theorem for Riemannian manifolds with a Killing field

The classical Bernstein theorem asserts that any complete minimal surface in Euclidean space that can be written as the graph of a function on must be a plane. In this paper, we extend Bernstein’s result to complete minimal surfaces in (may be non-complete) ambient spaces of non-negative Ricci curvature carrying a Killing field. This is done under the assumption that the sign of the angle function between a global Gauss map and the Killing field remains unchanged along the surface. In fact, our main result only requires the presence of a homothetic Killing field. L.J. Alías was partially supported by MEC/FEDER project MTM2004-04934-C04-02, F. Séneca project 00625/PI/04, and F. Séneca grant 01798/EE/05, Spain     

The free fermion field as a Markov field

A definition of a Markov field is given which allows for noncommuting fields. In the commutative case, we recover Nelson's definition (E. Nelson, Construction of quantum fields from Markoff fields, J. Functional Analysis12 (1973), 97–112). Conditional expectations are shown to exist in a regular probability gage space, and, using an independence property of these in the free fermion gage space, it is shown that the free fermion field over $\text{H}$^?1($\text{R}$^d) is a Markov field.     

Killing vector fields of constant length on Riemannian manifolds

We study the nontrivial Killing vector fields of constant length and the corresponding flows on complete smooth Riemannian manifolds. Various examples are constructed of the Killing vector fields of constant length generated by the isometric effective almost free but not free actions of S ¹ on the Riemannian manifolds close in some sense to symmetric spaces. The latter manifolds include “almost round” odd-dimensional spheres and unit vector bundles over Riemannian manifolds. We obtain some curvature constraints on the Riemannian manifolds admitting nontrivial Killing fields of constant length.     

Killing vector fields on Riemannian and Lorentzian 3-manifolds

We give a complete local classification of all Riemannian 3-manifolds $(M,g)$ admitting a nonvanishing Killing vector field T. We then extend this classification to timelike Killing vector fields on Lorentzian 3-manifolds, which are automatically nonvanishing. The two key ingredients needed in our classification are the scalar curvature S of g and the function $\text{Ric}(T,T)$, where Ric is the Ricci tensor; in fact their sum appears as the Gaussian curvature of the quotient metric obtained from the action of T. Our classification generalizes that of Sasakian structures, which is the special case when $\text{Ric}(T,T)=2$. We also give necessary, and separately, sufficient conditions, both expressed in terms of $\text{Ric}(T,T)$, for g to be locally conformally flat. We then move from the local to the global setting, and prove two results: in the event that T has unit length and the coordinates derived in our classification are globally defined on ${\mathbb{R}}^3$, we give conditions under which S completely determines when the metric will be geodesically complete. In the event that the 3-manifold M is compact, we give a condition stating when it admits a metric of constant positive sectional curvature.     

When is a vector field injective?

In this paper we define a notion of injectivity for a vector field over a Riemannian manifold and we give a sufficient condition for it. The proof is an extension of a well known Theorem stating that a map from an open convex set of the euclidean space into is a diffeomorphism onto its image provided that the bilinear operator associated to its differential is (positive or negative) definite everywhere. In the second part of the paper, we show that these results have a natural extension to the tangent bundle, with straightforward applications to second order differential equations. Received May 3, 1997 - Revised version received October 21, 1997