arXiv research

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…

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…

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

Paper compares five surface Navier-Stokes derivations and finds some are equivalent.

Study on a new class of meanders with tangential intersections.

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…

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…

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…

In this article, we study rectifying curves in arbitrary dimensional Euclidean space. A curve is said to be a rectifying curve if, in all points of the curve, the orthogonal complement of its normal vector contains a fixed point. We characterize rectifying curves in the $n$-dimensional Euclidean space in different ways…

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…

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 proves global existence and decay for complex wave equations.

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…

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

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.

New conditions ensure points can be uniquely represented by combinations of variety elements.

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

Proves stability of Minkowski space-time for Einstein-Yang-Mills equations.

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 …

Develops tensor calculus for submanifolds of arbitrary codimension.

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

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…

Finite element method approximates scalar curvature in arbitrary dimensions.

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.

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…

The paper approximates Einstein tensor using finite elements.

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 main result of the present paper is a coincidence formula for foliated manifolds. To prove this we establish Kuenneth formula, Poincare duality and intersection product in the context of tangential de Rham cohomology and homology of tangential currents. We apply the formula to get a dynamical Lefschetz formula for …

we discuss the decomposition of the zeta-determinant of the square of the Dirac operator into the contributions coming from the different parts of the manifold in the case of an invertible tangential operator.

Recent developments on biconservative submanifolds in Riemannian geometry.

Classical Ljusternik-Schnirelmann category is upper bounded by the number of critical points of any bounded from below differentiable functions of Palais-Smale type. Here we achieve an adaptation of this result for the tangential category of foliations. We introduce a weaker type of Palais-Smale function, obtaining a s…

Generalizes Poincaré-Hopf Theorem for piecewise smooth boundaries.

Under mild assumptions on a group G, we prove that the class of complete Riemannian n-manifolds of uniformly bounded negative sectional curvatures and with the fundamental groups isomorphic to G breaks into finitely many tangential homotopy types. It follows that many aspherical manifolds do not admit complete negative…

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

The paper studies curves of constant-ratio in pseudo-Galilean space.

The study extends classical results on harmonic functions to Riemannian manifolds with non-tangential boundary limits.

Proposes a new method for better explaining neural network decisions.

In this paper we analyze the tangential symmetries of Darboux integrable decomposable exterior differential systems. The decomposable systems generalize the notion of a hyperbolic exterior differential system and include the classic notion of Darboux integrability for first order systems and second order scalar equatio…

Let $R$ be a compact surface and let $Γ$ be a Jordan curve which separates $R$ into two connected components $Σ_1$ and $Σ_2$. A harmonic function $h_1$ on $Σ_1$ of bounded Dirichlet norm has boundary values $H$ in a certain conformally invariant non-tangential sense on $Γ$. We show that if $Γ$ is a quasicircle, then th…

Characterizes stable minimal capillary surfaces with specific angles.

We introduce and study co-dimension one area-minimizing locally rectifiable currents $T$ with $C^{1,α}$ tangentially immersed boundary: $\partial T$ is locally a finite sum of orientable co-dimension two submanifolds which only intersect tangentially with equal orientation. We show that any such $T$ is supported in a s…