For any ε > 0, we construct an explicit smooth Riemannian metric on the sphere Sn, n ≥ 3, that is within ε of the round metric and has a geodesic for which the corresponding orbit of the geodesic flow is ε-dense in the unit tangent bundle. Moreover, for any ε > 0, we construct a smooth Riemannian metric on S, n ≥ 3, that is within ε of the round metric and has a geodesic for which the complemen...

In this paper we study a Riemanian metric on the tangent bundle T (M) of a Riemannian manifold M which generalizes Sasaki metric and Cheeger Gromoll metric and a compatible almost complex structure which together with the metric confers to T (M) a structure of locally conformal almost Kählerian manifold. This is the natural generalization of the well known almost Kählerian structure on T (M). W...

In this note, we show that the holonomy group of a Riemannian connection on a k-dimensional Euclidean vector bundle is transitive on the unit sphere bundle whenever the Euler class χ is spherical. We extract several consequences from this, among them that this is always the case as long as χ does not vanish, and the base of the bundle is simply connected and rationally (k + 1)/2-connected.

We introduce the concepts of second-order radial composed tangent derivative, second-order radial tangent derivative, second-order lower radial composed tangent derivative, and second-order lower radial tangent derivative for set-valued maps by means of a radial tangent cone, second-order radial tangent set, lower radial tangent cone, and second-order lower radial tangent set, respectively. Som...

In this study, we give the relationships between the conical curvatures of ruled surfaces generated by the unit vectors of the ruling, central normal and central tangent of a ruled surface in the Euclidean 3-space E^3. We obtain differential equations characterizing slant ruled surfaces and if the reference ruled surface is a slant ruled surface, we give the conditions for the surfaces generate...