arXiv research

Proof of Tait-Kneser theorem and related variations using Lorentzian geometry.

Study of rigid body displacements in a projective space over dual numbers with geometric interpretations.

We discuss the theorem on the existence of six points on a convex closed plane curve in which the curve has a contact of order six with the osculating conic. (This is the ``projective version'' of the well known four vertices theorem for a curve in the Euclidean plane.) We obtain this classical fact as a corollary of s…

New surface class defined using osculating circles.

In this study, we define a new type of direction curves in the Euclidean 3-space such as osculating-direction curve. We give the characterizations for these curves. Moreover, we obtain the relationships between osculating direction curves and some special curves such as helix, slant helix or rectifying curves.

Study projective connections on surfaces using osculating spaces.

We view conformal surfaces in the 4--sphere as quaternionic holomorphic curves in quaternionic projective space. By constructing enveloping and osculating curves, we obtain new holomorphic curves in quaternionic projective space and thus new conformal surfaces. Applying these constructions to Willmore surfaces, we show…

The classical Tait-Kneser theorem states that the osculating circles of a smooth plane curve, free from curvature extrema, are pairwise disjoint. We prove a number of analogs of this theorem, e.g., for ovals of osculating cubics, osculating polynomials and trigonometric polynomials; in each case, we will obtain a non-d…

In an article of 1967 W. Edge gave a description of some beautiful geometric properties of the Kummer surface complete intersection of three quadrics in $\mathbb P^5$. Working on it, R. Dye proved that all its osculating spaces have dimension less than the expected 5. Here we discuss these results, also at the light of…

In this article we present a study of the subspaces of the manifold OscM, the total space of the osculator bundle of a real manifold M. We obtain the induced connections of the canonical metrical N-linear connection determined by the homogeneous prolongation of a Finsler metric to the manifold OscM. We present the rela…

The Tait-Kneser theorem states that the osculating circles of a plane curve with monotonic curvature are pairwise disjoint and nested. We discuss this theorem and a number of its variations.

Osculating spaces of decomposable scrolls (of any genus and not necessarily normal)are studied and their inflectional loci are related to those of their generating curves by using systematically an idea introduced by Piene and Sacchiero in the setting of rational normal scrolls. In this broader setting the extra compon…

Using the reviewed Riemann-Liouville fractional derivative we introduce the fractional osculator Lagrange space of k order and the main structures on it. The results are applied at the k order fractional prolongation of Lagrange, Finsler and Riemann fractional structures.

We study the Jacobi osculating rank of geodesics on naturally reductive homogeneous manifolds and we apply this theory to the 3-dimensional case. Here, each non-symmetric, simply connected naturally reductive 3-manifold can be given as a principal bundle over a surface of constant curvature, such that the curvature of …

In this paper, using the method of moving frames, we generalise some of Terracini's results on varieties with tangent defect. In particular, we characterise varieties with higher order osculating defect in terms of Jacobians of higher fundamental forms and moreover we characterise varieties with "small" higher fundamen…

This paper proves the manifold hypothesis for lower embedding dimensions using osculating hyperspheres.

This article is devoted to the study of cyclides osculating general surfaces. We show that generically, at any point of a surface, one has a one-parameter family of cyclides tangent to a surface curve of order three and among them just one is tangent to this curve of order four. This one will be called the osculating c…

We introduce a method to produce bounds for the non secant defectivity of an arbitrary irreducible projective variety, once we know how its osculating spaces behave in families and when the linear projections from them are generically finite. Then we analyze the relative dimension of osculating projections of Grassmann…

The paper glosses different forms of an introducing of higher order tangent-like functors, especially functors derived from higher order nonholonomic tangent functors. A special attention is devoted to higher order osculating bundles: their identification with higher order tangent bundles is demonstrated as the main re…

In this paper, we analyze the geometric structure of an Euclidean submanifold whose osculating spaces form a nonconstant family of proper subspaces of the same dimension. We prove that if the rate of change of the osculating spaces is small, then the submanifold must be a (submanifold of a) ruled submanifold of a very …

The set of osculating circles of a given curve in $\SS^3$ forms a curve in the set of oriented circles in $\SS^3$. We show that its "${\frac12}$-dimensional measure" with respect to the pseudo-Riemannian structure of the set of circles is proportional to the conformal arc-length of the original curve, which is a confor…

We explore the geometry that underlies the osculating nilpotent group structures of the Heisenberg calculus. For a smooth manifold $M$ with a distribution $H\subseteq TM$ analysts use explicit (and rather complicated) coordinate formulas to define the nilpotent groups that are central to the calculus. Our aim in this p…

We prove a generalization of the Monge-Cayley-Salmon theorem on osculation and ruled submanifolds using elementary geometric measure theory.

It is the purpose of the present paper to outline an introduction in theory of embeddings in the manifold Osc^{2}M. First, we recall the notion of 2-osculator bundle. The second section is dedicated to the notion of submanifold in the total space of the 2-osculator bundle, the manifold Osc^{2}M. A moving frame is const…

The "dancing metric" is a pseudo-riemannian metric $\pmb{g}$ of signature $(2,2)$ on the space $M^4$ of non-incident point-line pairs in the real projective plane $\mathbb{RP}^2$. The null-curves of $(M^4,\pmb{g})$ are given by the "dancing condition": the point is moving towards a point on the line, about which the li…

It is the purpose of the present paper to outline an introduction in theory of embeddings in the manifold Osc^{2}M. First, we recall the notion of 2-osculator bundle. The second section is dedicated to the notion of submanifold in the total space of the 2-osculator bundle, the manifold Osc^{2}M. A moving frame is const…

Several classes of irreducible orthogonal representations of compact Lie groups that are of importance in Differential Geometry have the property that the second osculating spaces of all of their nontrivial orbits coincide with the representation space. We say that representations with this property are of class O^2. O…

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…

The orthogonal trajectories of the first tangents of the curve are called the involutes of $x$. The hyperspheres which have higher order contact with a curve $x$ are known osculating hyperspheres of $x$. The centers of osculating hyperspheres form a curve which is called generalized evolute of the given curve $x$ in $n…

This is the appendix of the paper [T. Arias-Marco, Constant Jacobi osculating rank of $U(3)/(U(1) \times U(1) \times U(1))$, Arch. Math. (Brno) 45 (2009), 241--254] where we obtain an interesting relation between the covariant derivatives of the Jacobi operator valid for all geodesic on the flag manifold $M^6=U(3)/(U(1…

New type of ruled surfaces studied with properties and examples.

CMC-1 surfaces linked via Möbius transformations between circle patterns.

The paper characterizes curves in pseudo-Galilean 4-space.

We show that, for mechanical system with external forces, the equations of deviations of solution curves of the corresponding Lagrange equations,determine a nonlinear connection on the second order osculator (second order tangent) bundle. In particular, Jacobi equations in Finsler and Riemann spaces determine such a no…

The purpose of this paper is to generalize the convexity of Mabuchi's functional to the conic setting. We first established a frame to study conic cscK metrics, and then the conic Mabuchi functional was introduced in such a way that conic cscK metrics are its critical points. Finally we proved the convexity of the coni…

The paper classifies isoparametric hypersurfaces in conic Finsler spaces.

Two unique conic-line arrangements with degree 9 are found.

Study conic singular manifolds, proving Lipschitz normal embedding.

Desingularizes conically singular Cayley submanifolds.

It is well known that Cayley's ruled cubic surface carries a three-parameter family of twisted cubics sharing a common point, with the same tangent and the same osculating plane. We report on various results and open problems with respect to contact of higher order and dual contact of higher order for these curves.

The generic singularities and bifurcations are classified for one-parameter families of curves with frames in a space form, the Euclidean space, the elliptic space or the hyperbolic space via projective geometry. Two kinds of frames are considered, adapted frames and osculating frames, in terms of certain differential …

Proves existence of Yamabe metrics on conical manifolds with conical points and links.

Proves conditions for positive scalar curvature on certain manifolds with conical singularities.

Confocal conics form an orthogonal net. Supplementing this net with one of the following: 1) the net of Cartesian coordinate lines aligned along the principal axes of conics, 2) the net of Apollonian pencils of circles whose foci coincide with the foci of conics, 3) the net of tangents to a conic of the confocal family…

Study on spherical conical metrics and their reducibility on compact Riemann surfaces.

In this paper we develop a systematic deformation theory for conic constant curvature metrics on a closed surface when all cone angles are less than $2π$; in particular, we define and study the Teichmüller space $\mathcal{T}^{\mathrm{conic}}_{γ,k}$ of conic constant curvature metrics on a surface of genus $γ$ with $k$ …

Based on C. Li and Y. Rubinstein's upper bisectional curvature bound estimate for the conic Kähler metric, we can construct a smoothing sequence for the conic metric with uniformly upper bisectional curvature bound. For the conic metric along a simple normal crossing divisor with triple or higher multiple points we may…

Proves smoothness of conical singularities in mean curvature flow.