We investigate bi–Hamiltonian structures and related mKdV hierarchy of solitonic equations generated by (semi) Riemannian metrics and curve flow of non–stretching curves. The corresponding nonholonomic tangent space geometry is defined by canonically induced nonlinear connections, Sasaki type metrics and linear connections. One yields couples of generalized sine–Gordon equations when the corres...

In this paper, we deal with harmonic metrics respect to generalized Kantowski–Sachs type spacetime metrics. We also consider the Sasaki, horizontal and complete lifts of tangent bundle study their harmonicity.

We consider a complete nonsingular variety X over C, having a normal crossing divisor D such that the associated logarithmic tangent bundle is generated by its global sections. We show that H (

Multisoliton manifolds are characterized as symplectic prime ideals of the symplectic Lie algebra module generated by symmetries and mastersymmetries. This identification allows an explicit construction of the tangent bundle of the multisoliton manifolds.

The aim of the note is to give a complete description all hyperplane sections projective bundle associated tangent $${{{\mathbb {P}}}}^2$$ under its natural embedding in $${\mathbb {P}}^7.$$ As an application one obtains possible deformations homogeneous space $$\text {SL}_3({\mathbb {C}})/B$$ co-dimension sub-scheme which union two fundamental Schubert divisors.

The notion of an implicit Hamiltonian system—an isotropic mapping H : M → (TM, ω̇) into the tangent bundle endowed with the symplectic structure defined by canonical morphism between tangent and cotangent bundles of M—is studied. The corank one singularities of such systems are classified. Their transversality conditions in the 1-jet space of isotropic mappings are described and the correspondin...

We consider the evolution of a compact segment of an analytic curve on the unit tangent bundle of a finite volume hyperbolic n-manifold under the geodesic flow. Suppose that the curve is not contained in a stable leaf of the flow. It is shown that under the geodesic flow, the normalized parameter measure on the curve gets asymptotically equidistributed with respect to the normalized natural Rie...

An Engel structure is a maximally non-integrable field of two-planes tangent to a four-manifold. Any two Engel structures are locally diffeomorphic. We investigate the deformation space of Engel structures obtained by deforming certain canonical Engel structures on four-manifolds with boundary. When the manifold is RP 3 × I where I is a closed interval, we show that this deformation space conta...

At the heart of differential geometry is the construction of the tangent bundle of a manifold. There are various abstractions of this construction, and of particular interest here is that of Tangent Structures. Tangent Structure is defined via giving an underlying categoryM and a tangent functor T along with a list of natural transformations satisfying a set of axioms, then detailing the behavi...

In this article, we present some results concerning the harmonicity on tangent bundle equipped with vertical rescaled metric. We establish necessary and sufficient conditions under which a vector field is harmonic respect to metric construct examples of fields. also study along map between Riemannian manifolds, target manifold its bundle. Next discuss composition projection from into another ma...