In this paper, a notion of principal $2$-bundle over Lie groupoid has been introduced. For such $2$-bundles, we produced short exact sequence VB-groupoids, namely, the Atiyah sequence. Two notions connection structures viz. strict connections and semi-strict on arising respectively, from retraction up to natural isomorphism have We constructed class $\mathbb{G}=[G_1\rightrightarrows G_0]$-bundl...

We propose a new systematic fibre bundle formulation of nonrelativistic quantum mechanics. The new form of the theory is equivalent to the usual one but it is in harmony with the modern trends in theoretical physics and potentially admits new generalizations in different directions. In it a pure state of some quantum system is described by a state section (along paths) of a (Hilbert) fibre bund...

We show that if L is a codimension-one lamination in a finite volume hyperbolic 3-manifold such that the principal curvatures of each leaf of L are all in the interval (−δ, δ) for a fixed δ ∈ [0, 1) and no complimentary region of L is an interval bundle over a surface, then each boundary leaf of L has a nontrivial fundamental group. We also prove existence of a fixed constant δ0 > 0 such that i...

In this paper, we extend Sasaki metric for tangent bundle of a Riemannian manifold and Sasaki-Mok metric for the frame bundle of a Riemannian manifold [I] to the case of a semi-Riemannian vector bundle over a semi- Riemannian manifold. In fact, if E is a semi-Riemannian vector bundle over a semi-Riemannian manifold M, then by using an arbitrary (linear) connection on E, we can make E, as a...

The canonical reduction algorithm is applied to Maxwell and Yang-Mills equations considered as Hamiltonian systems on some fiber bundles with symplectic and connection structures. The minimum interaction principle proved to have geometric origin within the reduction method devised. 0. Preliminaries We begin by reviewing the backgrounds of the reduction theory subject to Hamiltonian systems with...

Let M be a compact connected special flat affine manifold without boundary equipped with a Gauduchon metric g and a covariant constant volume form. Let G be either a connected reductive complex linear algebraic group or the real locus of a split real form of a complex reductive group. We prove that a flat principal G–bundle EG over M admits a Hermitian–Einstein structure if and only if EG is po...

In this paper, we propose a new splitting algorithm for dynamical low-rank approximation motivated by the fibre bundle structure of set fixed rank matrices. We first introduce geometric description matrices which relies on natural parametrization More precisely, it is endowed with analytic principal bundle, an explicit local charts. For matrix differential equations, order numerical integrator ...

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...

Abstract In this continuation of [4], we investigate the deformations holomorphic Cartan geometries where underlying complex manifold is allowed to move. The space infinitesimal a flat geometry computed. We show that natural forgetful map, from principal bundle on topological manifold, an isomorphism.