Related papers: Higher order tangent bundles

The tangent bundle $T^kM$ of order $k$, of a smooth Banach manifold $M$ consists of all equivalent classes of curves that agree up to their accelerations of order $k$. In the previous work of the author he proved that $T^kM$, $1\leq k\leq…

The second order tangent bundle $T^{2}M$ of a smooth manifold $M$ consists of the equivalent classes of curves on $M$ that agree up to their acceleration. It is known that in the case of a finite $n$-dimensional manifold $M$, $T^{2}M$…

In this paper we develop the geometry of bounded Fr\'echet manifolds. We prove that a bounded Fr\'echet tangent bundle admits a vector bundle structure. But the second order tangent bundle $T^2M$ of a bounded Fr\'echet manifold $M$, becomes…

Given a vector bundle $A\to M$ we study the geometry of the graded manifolds $T^*[k]A[1]$, including their canonical symplectic structures, compatible Q-structures and Lagrangian Q-submanifolds. We relate these graded objects to classical…

In this paper for a given Banach, possibly infinite dimensional, manifold $M$ we focus on the geometry of its iterated tangent bundle $T^rM$, $r\in {\N}\cup\{\infty\}$. First we endow $T^rM$ with a canonical atlas using that of $M$. Then…

A Riemannian metric bundle G(M) is a fiber bundle over a smooth manifold M, whose fibers are the spaces of symmetric, positive-definite bilinear forms on the tangent spaces of M, which represent the Rieman?nian metrics. In this work, we aim…

We present bounds for the geometric degree of the tangent bundle and the tangential variety of a smooth affine algebraic variety $V$ in terms of the geometric degree of $V$. We first analyze the case of curves, showing an explicit relation…

Let M be an n-dimensional Riemannian manifold and TM its tangent bundle. The conformal and fiber preserving vector fields on TM have well-known physical interpretations and have been studied by physicists and geometricians. Here we define a…

In our previous papers [Far East Journal of Mathematical Sciences, 35 (2009), 211-223] and [International Journal of Pure and Applied Mathematics, 60 (2010), 15-24] we have developed the theory of Weil prolongation, Weil exponentiability…

We study the holomorphic vector bundles E over the twistor space Tw(M) of a compact simply connected hyperk\"ahler manifold $M$. We give a characterization of the semistability condition for E in terms of its restrictions to the holomorphic…

Jet manifolds and vector bundles allow one to employ tools of differential geometry to study differential equations, for example those arising as equations of motions in physics. They are necessary for a geometrical formulation of…

We determine the curvature equations of natural metrics on tangent bundles and radius r tangent sphere bundles S_rM of a Riemannian manifold M. A family of positive scalar curvature metrics on S_rM is found, for any M with bounded sectional…

Motivated by generalized geometry, we discuss differential geometric structures on the total space $\mathfrak{T}M$ of the bundle $TM\oplus T^*M$, where $M$ is a differentiable manifold; $\mathfrak{T}M$ is called a big-tangent manifold. The…

We prove vanishing results for the generalized Miller-Morita-Mumford classes of some smooth bundles whose fiber is a closed manifold that supports a nonpositively curved Riemannian metric. We also find, under some extra conditions, that the…

We review the theory of quaternionic Kahler and hyperkahler structures. Then we consider the tangent bundle of a Riemannian manifold M with a metric connection D (with torsion) and with its well estabilished canonical complex structure.…

Let E and F be vector bundles over a complex projective smooth curve X, and suppose that 0 -> E -> W -> F -> 0 is a nontrivial extension. Let G be a subbundle of F, and D an effective divisor on X. We give a criterion for the subsheaf G(-D)…

A natural explicit condition is given ensuring that an action of the multiplicative monoid of non-negative reals on a manifold F comes from homotheties of a vector bundle structure on F, or, equivalently, from an Euler vector field. This is…

For a Riemann surface $X$ and the moduli of regularly stable $G$-bundles $M$, there is a naturally occuring "$adjoint$" vector bundle over $X \times M$. One can take the determinant of this vector bundle with respect to the projection map…

We give a general description of the construction of weighted spherically symmetric metrics on vector bundle manifolds, i.e. the total space of a vector bundle $E\rightarrow M$, over a Riemannian manifold $M$, when $E$ is endowed with a…

Let G be a Lie goup, let M and N be smooth connected G-manifolds, let f be a smooth G-map from M to N, and let P denote the fiber of f. Given a closed and equivariantly closed relative 2-form for f with integral periods, we construct the…