2k-inner products and 2k-Riemannian metrics

The notion of 2k-inner product is introduced as a generalization of usual inner product and Q-inner product([4]-[8]). As a consequence, is defined the notion of 2k-normed space and some properties, e.g. uniformly convexity, Gâteaux differentiability and Riesz propriety of the dual, are given. Also, the notion of 2k-Riemannian metric is introduced

Slant curves and particles in three-dimensional warped products and their Lancret invariants

Slant curves are introduced in three-dimensional warped products with Euclidean factors; these curves are characterized through the scalar product between the normal at the curve and the vertical vector field and an important feature is that the case of constant Frenet curvatures implies a proper mean curvature vector field. A Lancret invariant is obtained and the Legendre curves are analyzed as particular case. An example of a slant curve is given for the exponential warping function; our example illustrates a proper (i.e. not reducible to the two dimensions) case of the Lancret Theorem of $3$-dimensional hyperbolic geometry. We point out an eventually relationship with the geometry of relativistic models. 10.1017/S000497271200080

RECURRENT METRICS IN THE GEOMETRY OF SECOND ORDER DIFFERENTIAL EQUATIONS

Abstract. Given a pair (semispray S, metric g) on a tangent bundle, the family of nonlinear connections N such that g is recurrent with respect to (S, N ) with a fixed recurrent factor is determined by using the Obata tensors. In particular, we obtain a characterization for a pair (N, g) to be recurrent as well as for the triple (S, c N , g), where c N is the canonical nonlinear connection of the semispray S. Also, the Weyl connection of conformal gauge theories is obtained as a particular case