arXiv subjects

Ahmed Gamal Shaltut

Publications and source records attributed to Ahmed Gamal Shaltut.

3 recordsLinked to original sources

arXiv2026

Inverse limits of locally convex Lie groupoids

We study strict inverse systems of locally convex Lie groupoids and Lie algebroids. We show that strict inverse limits of Lie algebroids and Lie groupoids inherit natural Lie structures and establish a natural isomorphism of functors $\varprojlim\circ\operatorname{Lie}\cong\operatorname{Lie}\circ\varprojlim$. We then consider inverse systems of integrable Banach-Lie algebroids and their source-simply connected levelwise integrations. We give conditions under which these integrations form a strict inverse system of Lie groupoids and hence their inverse limit integrates the inverse-limit algebroid. Under additional hypotheses on the source-fiber inverse sequence, we describe the obstruction to source-simply connectedness of the resulting integration in terms of the derived inverse limit $\varprojlim^{1}π_2$ of the second homotopy groups of the source fibers. We illustrate the theory with several examples: inverse limits of jet groupoids, current groupoids, and gauge groupoids, including the corresponding inverse-limit Atiyah algebroids and a source-simply connected inverse-limit integration arising from the quaternionic Hopf bundle.

math.DG↗

arXiv2026

A category of locally convex Lie algebroids

We study first-order locally convex Lie algebroids in the setting of Bastiani calculus. The first-order condition is automatic in finite dimensions, but is an additional regularity hypothesis for general locally convex vector bundles. Under this condition, we define sheaves of scalar-valued and vector-valued Lie algebroid forms as fiberwise continuous alternating maps with smooth local representatives. We define morphisms by requiring the induced pullback on inverse-image sheaves of scalar-valued forms to commute with the Lie algebroid differentials, and prove that first-order locally convex Lie algebroids form a category. We also study representations and the induced cohomology sheaves. We show that locally convex Lie groupoids have first-order Lie algebroids and that Lie groupoid morphisms induce morphisms in this category. As applications, we prove that the current algebroid associated with a first-order Banach Lie algebroid is a first-order Fréchet Lie algebroid, and we prove a Lie II theorem in the Banach setting: first-order morphisms between the Lie algebroids of Banach-Lie groupoids under source-connected and source-simply connected hypotheses integrate to unique Lie groupoid morphisms.

math.DG↗

arXiv2025

Invariance of subbundles and nonholonomic trajectories

In the comparison of nonholonomic mechanics and constrained variational mechanics, invariant affine subbundles arise in the determination of the initial conditions where the two methods yield the same trajectories. Motivated by this, differential conditions are considered for invariant affine subbundles as they arise in the comparison of nonholonomic and constrained variational mechanics. First of all, the formal integrability of the resulting linear partial differential equation is determined using Spencer cohomology. Second, iterative formulae are provided that permit the determination of the largest invariant affine subbundle invariant under an affine vector field. Finally, the problem of a disc rolling on an inclined plane with no-slip is considered as an example to illustrate the theory.

math.DG↗