Moments of Sectional Curvature

Gregor Weingart

Home/Archive/arXiv

arXiv · 1707.06369

Moments of Sectional Curvature

Gregor Weingart

View original source ↗Read source document ↗Export metadata ↓

Abstract

The sectional curvature of a compact Riemannian manifold M can be seen as a random variable on the Grassmann bundle of 2-planes in TM endowed with the Fubini-Study volume density. In this article we calculate the moments of this random variable by integrating suitable local Riemannian invariants and discuss the distribution of the sectional curvature of Riemannian products. Moreover we calculate the moments and the distribution of the sectional curvature for all compact symmetric spaces of rank 1 explicitly and derive a formula for the moments of general symmetric spaces. Interpolating the explicit values for the moments obtained we prove a weak version of the Hitchin-Thorpe Inequality.

Explore related subjects

math.DG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Gregor Weingart. 2017-07-20. Moments of Sectional Curvature. https://arxiv.org/abs/1707.06369

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↗