arXiv research

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

We give a unified method for the general equivalence problem of extrinsic geometry, on the basis of our formulation of a general extrinsic geometry as that of an osculating map $\varphi\colon (M,\mathfrak f) \to L/L^0 \subset \operatorname{Flag}(V,φ)$ from a filtered manifold $(M,\mathfrak f)$ to a homogeneous space $L…

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…

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.

In differential geometry of surfaces the Dirac operator appears intrinsically as a tool to address the immersion problem as well as in an extrinsic flavour (that comes with spin transformations to comformally transfrom immersions) and the two are naturally related. In this paper we consider a corresponding pair of disc…

Study improves understanding of submanifold reach in Riemannian geometry.

In this note we survey recent results on the extrinsic geometry of the Jacobian locus inside $\mathsf{A}_g$. We describe the second fundamental form of the Torelli map as a multiplication map, recall the relation between totally geodesic subvarieties and Hodge loci and survey various results related to totally geodesic…

Paper proves a Penrose inequality in extrinsic geometry.

Study k-folding map-germs to understand surface geometry.

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

For an embedded conformal hypersurface with boundary, we construct critical order local invariants and their canonically associated differential operators. These are obtained holographically in a construction that uses a singular Yamabe problem and a corresponding minimal hypersurface with boundary. They include an ext…

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

The paper explores geometric invariants of null hypersurfaces using Carrollian 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 ${\…

Lectures on surface evolution through singularities.

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

The study proves a theorem on Riemannian manifolds for wedge products of weakly convergent differential forms.

New operators and curvatures derived from embedded manifolds.

CMS formulation solves Poincare conjecture for all dimensions.

Defines constraint tensor for null hypersurfaces, providing explicit geometry.

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

This essay, an excerpt of the author's Ph.D. in Philosophy of mathematics (2012) thought of as being a companion to recent discoveries of new explicit Cartan geometry curvatures, analyzes how Gauss, after having devised the isometrically invariant character of curvature, struggled with elimination computations in order…

Proposes eDNNs and iDNNs for deep learning on manifolds.

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…

In this paper, geometric characterizations of conformally flat and radially flat hypersurfaces in $\mathbb{S}^n \times \mathbb{R}$ and $\mathbb{H}^n \times \mathbb{R}$ are given by means of their extrinsic geometry. Under suitable conditions on the shape operator, we classify conformally flat hypersurfaces in terms of …

We prove the following localized version of a classical ellipsoid characterization: Let $B\subset\mathbb R^3$ be convex body with a smooth strictly convex boundary and 0 in the interior, and suppose that there is an open set of planes through 0 such that all sections of $B$ by these planes are linearly equivalent. Then…

It is classically known that generic smooth maps of R^2 into R^3 admit only cross cap singularities. This suggests that the class of cross caps might be an important object in differential geometry. We show that the standard cross cap (u,uv,v^2) has non-trivial isometric deformations with infinite dimensional freedom. …

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.

Kernel methods have had great success in Statistics and Machine Learning. Despite their growing popularity, however, less effort has been drawn towards developing kernel based classification methods on Riemannian manifolds due to difficulty in dealing with non-Euclidean geometry. In this paper, motivated by the extrins…

Study of tractor bundles and spacelike immersions in Lorentzian manifolds.

New curvature measures for 4D manifolds with corners defined and related to Gauss-Bonnet.

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.

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…

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

Classifies polar actions on 3D homogeneous spaces.

We study data-driven representations for three-dimensional triangle meshes, which are one of the prevalent objects used to represent 3D geometry. Recent works have developed models that exploit the intrinsic geometry of manifolds and graphs, namely the Graph Neural Networks (GNNs) and its spectral variants, which learn…

Develops tensor calculus for submanifolds of arbitrary codimension.