Sub-Riemannian geometry and Lie groups. Part II. Curvature of metric spaces, coadjoint orbits and associated representations

Marius Buliga

Home/Archive/arXiv

arXiv · math/0407099

Sub-Riemannian geometry and Lie groups. Part II. Curvature of metric spaces, coadjoint orbits and associated representations

Marius Buliga

View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper is the third in a series dedicated to the fundamentals of sub-Riemannian geometry and its implications in Lie groups theory: "Sub-Riemannian geometry and Lie groups. Part I", math.MG/0210189, available at http://arxiv.org/abs/math.MG/0210189, and "Tangent bundles to sub-Riemannian groups", math.MG/0307342, available at http://arxiv.org/abs/math.MG/0307342 .

Explore related subjects

math.MG math.SG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Marius Buliga. 2004-07-07. Sub-Riemannian geometry and Lie groups. Part II. Curvature of metric spaces, coadjoint orbits and associated representations. https://arxiv.org/abs/math/0407099

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

KEEP EXPLORING

Related papers

arXiv2026

Strong Negative Type of Star-shaped Manifolds

Metric spaces of negative type (NT) and strong negative type (SNT) are fundamental in wide applications across computer science and statistics. It is well-known that spheres equipped with geodesic distance are of NT, and subsets of a sphere that contain at most one pair of antipodal points (e.g. an open hemisphere) are of SNT. In this paper, we extend these results to a class of star-shaped manifolds. By constructing a metric $d$ based on layer-wise radial projections, we prove that every star-shaped manifold, equipped with $d$, is of NT. Moreover, we prove that a subset of a star-shaped manifold is of SNT under certain injectivity and directional asymmetry conditions. In particular, for a subset of a single layer, it is of SNT if it contains exactly one point from each preimage of an open hemisphere (a fully directional asymmetry); moreover, it is of SNT if and only if it contains at most one opposite-directional pair. Finally, we prove that under similar conditions, SNT holds for almost and essentially star-shaped manifolds, and for those with incomplete radial coverage.

math.MG↗

arXiv2026

The dimension-free threshold for maximal sections of $\ell_p$

We prove the existence of a sharp threshold $p_0 \approx 26.265\ldots$, such that for every $n\in\mathbb{N}$ and $p\geq p_0$, the volume maximizing section of the unit ball of $\ell_p^n$ is the section with Ball's hyperplane $(\frac{e_1+e_2}{\sqrt{2}})^\perp$, whereas for $p\in(2,p_0)$ the asymptotic maximizer as the ambient dimension $n\to\infty$ is the hyperplane orthogonal to the diagonal vector $\frac{e_1+\cdots+e_n}{\sqrt{n}}$.

math.MG↗

arXiv2026

Towards Hodge-Riemann relations for non-Archimedean analogs of valuations on convex sets

In [8], a non-Archimedean analogue of the space of translation-invariant even valuations on convex sets was introduced. In [7], motivated by a further analogy with the classical theory, this space was equipped with two multiplicative structures, the product and the convolution. Both structures satisfy Poincare duality and the (non-mixed) hard Lefschetz theorem. In this paper, we formulate a conjecture concerning a more general mixed versions of the hard Lefschetz theorem and the Hodge-Riemann relations. We prove the non-mixed Hodge-Riemann relations in degree 1 for the product and, equivalently, in codegree 1 for the convolution.

math.MG↗