Let $M$ be an orientable hypersurface in the Euclidean space $R^{2n}$ with induced metric $g$ and $TM$ be its tangent bundle. It is known that the tangent bundle $TM$ has induced metric $overline{g}$ as submanifold of the Euclidean space $R^{4n}$ which is not a natural metric in the sense that the submersion $pi :(TM,overline{g})rightarrow (M,g)$ is not the Riemannian submersion. In this paper...

We characterize the natural diagonal almost product (locally product) structures on the tangent bundle of a Riemannian manifold. We obtain the conditions under which the tangent bundle endowed with the determined structure and with a metric of natural diagonal lift type is a Riemannian almost product (locally product) manifold, or an (almost) para-Hermitian manifold. We find the natural diagona...

in this paper, we are going to study the g-natural metrics on the tangent bundle of finslermanifolds. we concentrate on the complex and kählerian and hermitian structures associated with finslermanifolds via g-natural metrics. we prove that the almost complex structure induced by this metric is acomplex structure on tangent bundle if and only if the finsler metric is of scalar flag curvature. t...

In this article, we generalize the concept of the Lie algebra of vector fields to the set of smooth sections of a T-bundle which is by definition a canonical generalization of the concept of a tangent bundle. We define a Lie bracket multiplication on this set so that it becomes a Lie algebra. In the particular case of tangent bundles this Lie algebra coincides with the Lie algebra of vector fie...

a module is said to be $pi$-extending provided that every projection invariant submodule is essential in a direct summand of the module. in this paper, we focus on direct summands and indecomposable decompositions of $pi$-extending modules. to this end, we provide several counter examples including the tangent bundles of complex spheres of dimensions bigger than or equal to 5 and certain hyper ...

Totally geodesic property of the Hopf vector field. Abstract We prove that the Hopf vector field is a unique one among geodesic unit vector fields on spheres such that the submanifold generated by the field is totally geodesic in the unit tangent bundle with Sasaki metric. As application, we give a new proof of stability (instability) of the Hopf vector field with respect to volume variation us...

1. Write the formula for the Liouville measure Λ on the unit tangent bundle T 1 H n. Prove that Λ is invariant under the actions by Isom(H n) and the geodesic flow g t on T 1 H n. 2. Let δ be a geodesic in the hyperbolic space H n and let π : H n → δ denote the nearest point projection map. Suppose that for some x, y ∈ H n and some r > 0 we have d(x, δ), d(y, δ) ≥ r. Prove that d(π(x), π(y)) ≤ ...

1. Write the formula for the Liouville measure Λ on the unit tangent bundle T 1 H n. Prove that Λ is invariant under the actions by Isom(H n) and the geodesic flow g t on T 1 H n. 2. Let δ be a geodesic in the hyperbolic space H n and let π : H n → δ denote the nearest point projection map. Suppose that for some x, y ∈ H n and some r > 0 we have d(x, δ), d(y, δ) ≥ r. Prove that d(π(x), π(y)) ≤ ...

An explicit construction of a geodesic flow-invariant distribution lying in the discrete series of weight 2k isotopic component is found, using techniques from representation theory of SL2(R). It is found that the distribution represents an AC measure on the unit tangent bundle of the hyperbolic plane minus an explicit singular set. Finally, via an averaging argument, a geodesic flow-invariant ...