arXiv research

In this article we introduce a generalization of the Newton transformation to the case of a system of endomorphisms. We show that it can be used in the context of extrinsic geometry of foliations and distributions yielding new integral formulas containing generalized extrinsic curvatures.

Study improves understanding of submanifold reach in Riemannian geometry.

New systems of linear PDEs discovered in 3D contact manifolds.

Paper proves a Penrose inequality in extrinsic geometry.

In the paper we prove, that extrinsic curvature does not impose restrictions on the topology of a contact structure, except the obvious ones.

Unified method for solving extrinsic geometry problems.

Study on ratio of intrinsic to extrinsic metrics and its relation to surface area.

The paper characterizes hypersurfaces in curved spaces using their geometry.

Extrinsic Geometric Flow (EGF) for a codimension-one foliation has been recently introduced by authors as deformations of Riemannian metrics subject to quantities expressed in terms of its second fundamental form. In the paper we introduce soliton solutions to EGF and study their geometry for totally umbilical foliatio…

The study defines and characterizes extrinsic catenaries in hyperbolic space.

We use the conformal invariance and the holographic correspondence to fully specify the dependence of entanglement entropy on the extrinsic geometry of the 2d surface $Σ$ that separates two subsystems of quantum strongly coupled ${\mathcal{N}}=4$ SU(N) superconformal gauge theory. We extend this result and calculate en…

This paper is a survey of some of the developments in coarse extrinsic geometry since its inception in the work of Gromov. Distortion, as measured by comparing the diameter of balls relative to different metrics, can be regarded as one of the simplist extrinsic notions. Results and examples concerning distorted subgrou…

New conditions for calibrated submanifolds in Riemannian geometry.

Paper proves Hamilton's pinching theorem using mean curvature flow.

New CRB derived for curved models using extrinsic geometry.

Study of scattering on singular Yamabe spaces using asymptotically hyperbolic manifolds.

Study reveals how manifold geometry impacts linear regression solutions.

Geometrically refines Cramér-Rao bound using extrinsic manifold curvature.

We develop variation formulas for the quantities of extrinsic geometry for adapted variations of metrics on almost-product (e.g. foliated) Riemannian manifolds, and apply them to study the total mixed scalar curvature of a distribution -- analogue of the classical Einstein-Hilbert action. The mixed scalar curvature ${\…

Surface Networks enhance GNNs with extrinsic geometry for 3D mesh modeling.

Study shows one-dimensional location-scale-shape models are flat in Wasserstein geometry.

Study on biconservative surfaces in 3D space forms, focusing on complete surfaces.

Defines constraint tensor for null hypersurfaces, providing explicit geometry.

Geodesic spheres are the only quasicomplete surfaces in 3-space-forms.

Proposes eDNNs and iDNNs for deep learning on manifolds.

We prove a general extrinsic rigidity theorem for homogeneous varieties in $\mathbb{CP}^N$. The theorem is used to show that the adjoint variety of a complex simple Lie algebra $\mathfrak{g}$ (the unique minimal G orbit in $\mathbb{P}\mathfrak{g}$) is extrinsically rigid to third order. In contrast, we show that the ad…

Study of how small changes in triangulated surfaces affect their geometry.

For a generic embedding of a smooth closed surface $M$ into $\mathbb R^4$, the subset of $\mathbb R^4$ which is the affine $λ$-equidistant of $M$ appears as the discriminant set of a stable mapping $M \times M \to \mathbb R^4$, hence their stable singularities are $A_k, \, k=2, 3, 4,$ and $C_{2,2}^{\pm}$. In this paper…

Abstract: Develops universal formulae for Q-curvatures and related operators in conformal geometry.

Study variational formulas for distribution geometry, finding critical metrics.

New kernel method for shape classification on Kendall shape space.

On real hypersurfaces in complex space forms many results are proven. In this paper we generalize some results concerning extrinsic geometry of real hypersurfaces, to CR submanifolds of maximal CR dimension in complex space forms.

Quaternionic reformulation simplifies surface curvature theory.

Study of tractor bundles and spacelike immersions in Lorentzian manifolds.

The paper explores discrete versions of the Dirac operator on polygonal surfaces.

Study relationships between intrinsic and extrinsic invariants of Riemannian almost k-product manifolds.

We extend the results given by Colbois, Dryden and El Soufi on the relationships between the eigenvalues of the Laplacian and an extrinsic invariant called intersection index, in two directions. First, we replace this intersection index by invariants of the same nature which are stable under small perturbations. Second…

We are interested in the local extrinsic geometry of smooth surfaces in 4-space, and classify jets of Monge forms by projective transformations according to $\mathcal{A}^3$-types of their central projections.

Study curves in a spacelike hypersurface using geometric and singularity theory.

Develops tensor calculus for submanifolds of arbitrary codimension.

Classifies polar actions on 3D homogeneous spaces.

The paper refines classical covariance asymptotics using geometric information geometry.

Study non-degenerate anisocurved surfaces in homogeneous 3-manifolds.

We study the geometry of a codimension-one foliation with a time-dependent Riemannian metric. The work begins with formulae concerning deformations of geometric quantities as the Riemannian metric varies along the leaves of the foliation. Then the Extrinsic Geometric Flow depending on the second fundamental form of the…

Explicit formulas for extrinsic Paneitz operators and Q-curvatures for totally umbilic hypersurfaces.

We produce examples of codimension one foliations of the Euclidean and hyperbolic planes with bounded geometry which are topologically products, but for which leaves are non-recursively distorted. That is, the function which compares intrinsic distances in leaves with extrinsic distances in the ambient space grows fast…

Upper bounds for eigenvalues on submanifolds in weighted manifolds.

A conformal structure on a manifold $M^n$ induces natural second order conformally invariant operators, called Möbius and Laplace structures, acting on specific weight bundles of $M$, provided that $n\ge 3$. By extending the notions of Möbius and Laplace structures to the case of surfaces and curves, we develop here th…