The diffential geometry of composition sequences of differentiable manifolds

A. Tsemo

Home/Archive/arXiv

arXiv · math/0210102

The diffential geometry of composition sequences of differentiable manifolds

A. Tsemo

View original source ↗Read source document ↗Export metadata ↓

Abstract

Let F_0=B,...,F_n be a sequence of differentiable manifolds, G_i a Lie subgroup of diffeomorphisms of F_i, and H_i a subgroup of G_i central in G_i. We suppose also given a locally trivial bundle p_{K_i} over F_{i-1} which typical fiber is K_i the quotient of G_i by H_i. The aim of this paper is to study the differential geometry of the following problem: classify sequences M_n\to...M_1, where each map from M_i to M_{i-1} is a locally trivial fibration which typical fiber is F_i and which transition functions image are elements of G_i. We associate to this problem a tower of gerbes and define for it the notion of connective structure, curvature and holonomy using the notion of free transitive distribution (free TD)

Explore related subjects

math.DG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

A. Tsemo. 2002-11-04. The diffential geometry of composition sequences of differentiable manifolds. https://arxiv.org/abs/math/0210102

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Cohomology of Lie algebroids over topological ringed spaces

We consider Lie algebroids over a topological ringed space as quasicoherent sheaves of Lie-Rinehart algebras. We express hypercohomology for a locally free Lie algebroid (not necessarily of finite rank) as a derived functor, and simplify it via Čech cohomology. Furthermore, we define the Hochschild hypercohomology of a sheaf of generalized bialgebras (using a derived functor) and study the cases of the universal enveloping algebroid and of the jet algebroid of a Lie algebroid. In the sequel, we present a version of Hochschild-Kostant-Rosenberg theorem for a locally free Lie algebroid, as well as its dual version.

math.DG↗

arXiv2026

Family index for Fredholm extensions of semi-Fredholm operators

This paper is devoted to an abstract analogue of elliptic boundary value problems, namely, Fredholm realizations of semi-Fredholm operators in a Hilbert space. Such a realization is determined by an abstract boundary condition, which is a subspace in the space of abstract boundary values. We find the $K^0$ index of a family of such abstract boundary value problems, or the $K^1$ index in the self-adjoint case, in terms of the corresponding family of abstract boundary conditions. Our approach is based on passing from a Fredholm operator to its graph. The graph forms a Fredholm pair with the horizontal subspace, and we prove the index formula by deforming the horizontal subspace instead of the operator.

math.DG↗

arXiv2026

Classifying Slice-Regular Polynomials via Group Actions on the Twistor Space

We study the equivalence classes of slice-regular functions $f:Ω\to\mathbb{H}$ on a symmetric slice domain $Ω$, and of their subclass made of polynomial slice-regular functions, with respect to the natural action of $\mathrm{PGL}(2,\mathbb{H})$ and its subgroups, by employing the twistor construction. In particular, we characterize slice--regular functions whose twistor lift is planar and belongs to a given orbit, and we find normal classes of slice-regular polynomials with respect to the action of a parabolic subgroup of $\mathrm{GL}(2,\mathbb{H})$.

math.DG↗