arXiv research

Defines observer-invariant time derivatives on moving surfaces.

Generalizes Poincaré-Hopf Theorem for piecewise smooth boundaries.

A submanifold is said to be tangentially biharmonic if the bitension field of the isometric immersion that defines the submanifold has vanishing tangential component. The purpose of this paper is to prove that a surface in Euclidean $3$-space has tangentially biharmonic normal bundle if and only if it is either minimal…

This paper solves a Calderón problem for Beltrami fields on manifolds.

We present a method to develop a Hodge theory for tangential cohomology of foliations by mimicing Witten's approach to ordinary Morse theory by perturbations of the Laplacian

The notion of Lagrangian $H$-umbilical submanifolds was introduced by B. Y. Chen in 1997, and these submanifolds have appeared in several important problems in the study of Lagrangian submanifolds from the Riemannian geometric point of view. Recently, the author introduced the notion of tangentially biharmonic submanif…

Study contact structures on projective spaces, proving infinite non-isotopic structures.

Gradient flows for surface energies with tensor fields are derived and analyzed.

For a compact, connected, oriented Riemannian $3$-manifold $(M, g)$ with smooth boundary $\partial M$, we explicitly give a local representation and a full symbol expression for the electromagnetic Dirichlet-to-Neumann map by factorizing Maxwell's equations and using an isometric transform. We prove that one can recons…

Poincaré-Hopf theorem extended to Filippov vector fields on 2D manifolds.

Paper finds obstructions to compact Clifford-Klein forms for tangential symmetric spaces.

The characterization of the Nambu-Poisson n-tensors as a subfamily of the Generalized-Poisson ones recently introduced (and here extended to the odd order case) is discussed. The homology and cohomology complexes of both structures are compared, and some physical considerations are made.

Algorithm finds real-analytic Legendrian representatives for every link type.

In this paper we initiate the study of Yamabe and quasi-Yamabe solitons on Euclidean submanifolds whose soliton fields are the tangential components of their position vector fields. Several fundamental results of such solitons were proved. In particular, we classify such Yamabe and quasi-Yamabe solitons on Euclidean hy…

Tangential families are 1-parameter families of rays emanating tangentially from smooth curves. We classify tangential family germs up to Left-Right equivalence: we prove that there are two infinite series and four sporadic simple singularities of tangential family germs (in addition to two stable singularities). We gi…

We study the tangential Poisson cohomology (TP-cohomology) of regular Poisson manifolds, first defined by Lichnerowicz using contravariant tensor fields. We show that for a regular Poisson manifold M, the TP-cohomology coincides with the leafwise de Rham (or Cech) cohomology of the symplectic foliation of M. Its comput…

We study tangential families, i.e. systems of rays emanating tangentially from given curves. We classify, up to Left-Right equivalence, stable singularities of tangential family germs (under deformations among tangential families) and we study their envelopes. We discuss applications of our results to the case of tange…

Study on properties of tangential hypersurfaces in product-like manifolds.

An n-dimensional submanifold X of a projective space P^N (C) is called tangentially degenerate if the rank of its Gauss mapping γ: X ---> G (n, N) satisfies 0 < rank γ< n. The authors systematically study the geometry of tangentially degenerate submanifolds of a projective space $P^N (\mathbf{C})$. By means of the foca…

Our objective is to develop a stratified Morse theory with tangential conditions. We define a continuous strata-wise smooth Morse function on an abstract stratified space by using control conditions and radiality assumptions on the gradient vector field. For critical points of a Morse function one can show that the loc…

For a submanifold M in a Euclidean space, the tangential component x^T of the position vector field x of M is the most natural vector field tangent to the Euclidean submanifold, called the canonical vector field of M. In this article, first we prove that the canonical vector field of every Euclidean submanifold is alwa…

Investigates polar tangential angles of curves and their monotonicity.

Study examines tangential real hypersurfaces on Hermite-like manifolds.

Unified method visualizes curvature on curves and surfaces.

Study properties of specific solitons on submanifolds with special vector fields.

The position vector field x is the most elementary and natural geometric object on a Euclidean submanifold $M$. The position vector field plays very important roles in mathematics as well as in physics. Similarly, the tangential component x^T of the position vector field is the most natural vector field tangent to the …

A space curve in a Euclidean 3-space $\mathbb E^3$ is called a rectifying curve if its position vector field always lies in its rectifying plane. This notion of rectifying curves was introduced by the author in [Amer. Math. Monthly {\bf 110} (2003), no. 2, 147-152]. In this present article, we introduce and study the n…

The paper approximates Levi-Civita connection and curvature on 2D manifolds using finite elements.

Let $(φ_t)$ be a holomorphic semigroup of the unit disc (i.e., the flow of a semicomplete holomorphic vector field) without fixed points in the unit disc and let $Ω$ be the starlike at infinity domain image of the Koenigs function of $(φ_t)$. In this paper we completely characterize the type of convergence of the orbit…

We construct a Legendrian version of Envelope theory. A tangential family is a 1-parameter family of rays emanating tangentially from a smooth plane curve. The Legendrian graph of the family is the union of the Legendrian lifts of the family curves in the projectivized cotangent bundle $PT^*\R^2$. We study the singular…

Derives Lagrangian for minimal surfaces, proving tangential variations vanish.

In this paper we study tangentially degeneracy of the orbits of s-representations in the sphere. We show that an orbit of an s-representation is tangentially degenerate if and only if it is through a long root, or a short root of restricted root system of type G_2. Moreover these orbits provide many new examples of tan…

The tangential map is a map on the set of smooth planar curves. It satisfies the 3D-consistency property and is closely related to some well-known integrable equations.

We study invariant submanifolds of manifolds endowed with a normal or complex metric contact pair with decomposable endomorphism field $φ$. For the normal case, we prove that a $φ$-invariant submanifold tangent to a Reeb vector field and orthogonal to the other one is minimal. For a $φ$-invariant submanifold $N$ everyw…

We etablish a necessary and sufficient condition under which there exists a tangential and well graded star product, differential or not, on the dual g^* of a nilpotent Lie algebra g. We also give enlightening examples with explicit computations.

We obtain very sharp results about the lack of validity of the Poincare lemma for the tangential Cauchy Riemann equations, acting on tangential forms, tangential to a CR manifold M of general CR dimension n, and general CR codimension k. This generalizes the classical nonsolvability example of H. Lewy. We also discuss …

Study biharmonic submanifolds in warped product structures.

New method for mesh denoising using TGV of normal vector field.

We give a global description of envelopes of geodesic tangents of regular curves in (not necessarily convex) Riemannian surfaces. We prove that such an envelope is the union of the curve itself, its inflectional geodesics and its tangential caustics (formed by the conjugate points to those of the initial curve along th…

Study active nematic forces on curved surfaces, revealing new coupling mechanisms.

The paper analyzes thin-shell limits for viscous operators on Riemannian hypersurfaces.

A point $p\in\mathbb{P}^N$ of a projective space is $h$-identifiable, with respect to a variety $X\subset\mathbb{P}^N$, if it can be written as linear combination of $h$ elements of $X$ in a unique way. Identifiability is implied by conditions on the contact locus in $X$ of general linear spaces called non weak defecti…

Develops tensor calculus for submanifolds of arbitrary codimension.

We construct a tangential map from a locally symmetric space of noncompact type to its dual compact type twin. By comparing the induced map in cohomology to a map defined by Matsushima, we conclude that in the equal rank case the map has a nonzero degree.

New algorithms solve tensor problems with random components using SDP.

Study connections on Lie groupoids and stacks using Atiyah sequences.

We study the local differential geometry of varieties $X^n\subset \Bbb C\Bbb P^{n+a}$ with degenerate secant and tangential varieties. We show that the second fundamental form of a smooth variety with degenerate tangential variety is subject to certain rank restrictions. The rank restrictions imply a slightly refined v…

Solves characteristic problem in general relativity for null data.