Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0708.1100

Differential geometry of curves in Lagrange Grassmannians with given Young diagram

Igor Zelenko·Chengbo Li
View original source ↗Read source document ↗Export metadata ↓

Abstract

Curves in Lagrange Grassmannians appear naturally in the intrinsic study of geometric structures on manifolds. By a smooth geometric structure on a manifold we mean any submanifold of its tangent bundle, transversal to the fibers. One can consider the time-optimal problem naturally associate with a geometric structure. The Pontryagin extremals of this optimal problem are integral curves of certain Hamiltonian system in the cotangent bundle. The dynamics of the fibers of the cotangent bundle w.r.t. this system along an extremal is described by certain curve in a Lagrange Grassmannian, called Jacobi curve of the extremal. Any symplectic invariant of the Jacobi curves produces the invariant of the original geometric structure. The basic characteristic of a curve in a Lagrange Grassmannian is its Young diagram. The number of boxes in its kth column is equal to the rank of the kth derivative of the curve (which is an appropriately defined linear mapping) at a generic point. We will describe the construction of the complete system of symplectic invariants for parameterized curves in a Lagrange Grassmannian with given Young diagram. It allows to develop in a unified way local differential geometry of very wide classes of geometric structures on manifolds, including both classical geometric structures such as Riemannian and Finslerian structures and less classical ones such as sub-Riemannian and sub-Finslerian structures, defined on nonholonomic distributions.

Explore related subjects

math.DG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Igor Zelenko, Chengbo Li. 2007-08-08. Differential geometry of curves in Lagrange Grassmannians with given Young diagram. https://arxiv.org/abs/0708.1100

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

KEEP EXPLORING

Related papers

arXiv2026

A duality theorem for a four dimensional Willmore energy

We prove an analogue of the energy identities underlying Bryant's duality theorem for a four-dimensional Willmore energy $\mathcal{E}_{\rm GR}$ obtained by Graham--Reichert and Zhang in codimension one. We show that, for an immersion $Φ$ of a compact four-dimensional manifold without boundary $Σ$ into $\mathbb{R}^5$, the energy $\mathcal{E}_{\rm GR}(Φ)$ is equal to two energies associated with its conformal Gauss map $Y$: one defined only in terms of the image of $Y$, which is the analogue of the area functional for Willmore surfaces, and another defined on maps from $Σ$ into the de Sitter space $\mathbb{S}^{5,1}$, which is the analogue of the Dirichlet energy for Willmore surfaces. We prove that, even when restricted to immersions of a given topological manifold $Σ^4$, $\mathcal{E}_{\rm GR}$ is not bounded below on the set of immersions from $Σ$ into $\mathbb{R}^5$. We exhibit a second conformally invariant energy $\mathcal{E}_{\rm P}$ that is bounded below and whose construction is closer to that of the two-dimensional Willmore energy.

math.DG↗
arXiv2026

Barycentric subspace analysis of network-valued data

Certain data are naturally modeled by networks or weighted graphs, be they biological networks or mobility networks. When there is no canonical labeling of the nodes across the dataset, we talk about unlabeled networks. In this paper, we focus on the question of exploratory analysis of this type of data. More specifically, we address the issue of interpreting the feature subspace constructed by dimensionality reduction methods. Most existing methods for network-valued data are derived from principal component analysis (PCA) and therefore rely on subspaces generated by a set of vectors, which we identify as a major limitation in terms of interpretability. Instead, we propose to implement the method called barycentric subspace analysis (BSA), which relies on subspaces generated by a set of points, which we choose, in practice, to be samples. In order to provide a computationally feasible framework for BSA, we introduce a novel embedding for unlabeled networks where we replace their usual representation by equivalence classes of isomorphic networks with that by equivalence classes of cospectral networks. In a simulated study, we demonstrate the improved interpretability of BSA compared to tangent PCA. We then illustrate through two real-world datasets how BSA can be used both to visualize known patterns and to discover new ones in network-valued distributions.

math.DG↗
arXiv2026

Isoparametric Hypersurfaces in Products of Simply Connected Space Forms

For $i\in\{1,2\}, $ let $\mathbb Q_{ε_i}^{n_i}$ denote the simply connected space form of dimension $n_i\ge 2$ and constant sectional curvature $ε_i\in\mathbb R$, $(ε_1,ε_2)\ne (0,0)$. We prove that any connected isoparametric hypersurface of $\mathbb Q_{ε_1}^{n_1}\times\mathbb Q_{ε_2}^{n_2}$ has constant angle function. We use this property to classify both the isoparametric and homogeneous hypersurfaces of $\mathbb Q_{ε_1}^{n_1}\times\mathbb Q_{ε_2}^{n_2}$ when $ε_1ε_2\le 0$. Under additional hypotheses, we obtain a similar classification result in the case $ε_1ε_2>0$.

math.DG↗