arXiv research

This paper finds all prime alternating knots with minimal warping degree two.

The study finds lower bounds for the warping degree of a knot projection.

For an oriented link diagram D, the warping degree d(D) is the smallest number of crossing changes which are needed to obtain a monotone diagram from D. We show that d(D)+d(-D)+sr(D) is less than or equal to the crossing number of D, where -D denotes the inverse of D and sr(D) denotes the number of components which hav…

We introduce the warping matrix which is a new description of oriented knots from a viewpoint of warping degree.

Innovative warping labeling for twisted knots and braids.

Paper studies Hessian quotient equations in warped product manifolds.

Existence of hypersurfaces in warped product manifolds proven.

The warping matrix has been defined for knot projections and knot diagrams by using warping degrees. In particular, the warping matrix of a knot diagram represents the knot diagram uniquely. In this paper we show that the rank of the warping matrix is one greater than the crossing number. We also discuss the linearly i…

Study on bounds of knot untangling for specific types of knots.

The warping sum $e(K)$ of a knot $K$ is the minimal value of the sum of the warping degrees of a minimal diagram of $K$ with both orientations. In this paper, knots $K$ with $e(K) \le 3$ are characterized, and some knots $K$ with $e(K)=4$ are given.

For an oriented knot diagram D, the warping degree d(D) is the smallest number of crossing changes which are needed to obtain the monotone diagram from D in the usual way. We show that d(D) + d(-D) + 1 is less than or equal to the crossing number of D. Moreover the equality holds if and only if D is an alternating diag…

The study proves uniqueness and symmetry of self-similar solutions in warped product spaces.

The paper studies unknotting operations and numbers for plus-welded knotoids.

We introduce the warping crossing polynomial of an oriented knot diagram by using the warping degrees of crossing points of the diagram. Given a closed transversely intersected plane curve, we consider oriented knot diagrams obtained from the plane curve as states to take the sum of the warping crossing polynomials for…

We consider a family of manifolds with a class of degenerating warped product metrics $g_ε=ρ(ε,t)^{2a}dt^2 +ρ(ε,t)^{2b}ds_M^2$, with $M$ compact, $ρ$ homogeneous degree one, $a \le -1$ and $b > 0$. We study the Laplace operator acting on $L^{2}$ differential $p$-forms and give sharp accumulation rates for eigenvalues n…

Characterizes warping functions in Einstein Poisson warped spaces.

The paper defines the OU matrix for braid diagrams and finds determinant relationships.

The paper examines Einstein doubly warped product manifolds with a semi-symmetric metric connection.

Generalizes warped product submersion to conformal case.

The paper explores geometric properties of Riemannian warped product maps and their curvature.

Study shows how certain metrics can be split into warped products.

The goal of dynamic time warping is to transform or warp time in order to approximately align two signals together. We pose the choice of warping function as an optimization problem with several terms in the objective. The first term measures the misalignment of the time-warped signals. Two additional regularization te…

The paper proves rigidity for submanifolds in warped product manifolds.

Study on warped product Yamabe solitons with constant fiber curvature.

The paper explores warped-like product metrics with exceptional holonomy groups.

We prove that complete warped product Einstein metrics with isometric bases, simply connected space form fibers, and the same Ricci curvature and dimension are isometric. In the compact case we also prove that the warping functions must be the same up to scaling, while in the non-compact case there are simple examples …

Warped product affects divergences in information geometry.

Study properties of bi-warped product submanifolds in specific geometric spaces.

Study new warped metrics with vanishing Douglas curvature.

New rigidity found for 3D warped product domains.

The paper modifies a warped product space to find conditions for constant height functions.

Paper defines and studies Clairaut warped product Riemannian maps.

Study Einstein warped products with Einstein base and fiber.

Conditions for flat 3-manifolds with diagonal metrics are identified.

The warped product $N_1\times_f N_2$ of two Riemannian manifolds $(N_1,g_1)$ and $(N_2,g_2)$ is the product manifold $N_1\times N_2$ equipped with the warped product metric $g=g_1+f^2 g_2$, where $f$ is a positive function on $N_1$. Warped products play very important roles in differential geometry as well as in physic…

In this paper, we generalize the geometry of the product pseudo-Riemannian manifold equipped with the product Poisson structure (\cite{Nas2}) to the geometry of a warped product of pseudo-Riemannian manifolds equipped with a warped Poisson structure. We construct three bivector fields on a product manifold and show tha…

Characterizes and examines gradient solitons on doubly warped product manifolds.

In this paper we study fundamental geometric properties of doubly warped product immersion which is an extension of warped product immersion. Moreover, we study geometric inequality for doubly warped products isometrically immersed in arbitrary Riemannian manifolds.

In this paper we study the space of solutions to an overdetermined linear system involving the Hessian of functions. We show that if the solution space has dimension greater than one, then the underlying manifold has a very rigid warped product structure. We obtain a uniqueness result for prescribing the Ricci curvatur…

We derive one unified formula for Ricci curvature tensor on arbitrary warped product manifold by introducing a new notation for the lift vector and the Levi-Civita connection.This formula is helpful to further consider Ricci flow (RF) and hyperbolic geometric flow (HGF) and evolution equations on warped product manifol…

Study investigates concircular curvature on warped products.

In this paper, we study the Einstein warped products and multiply warped products with a quarter-symmetric connection. We also study warped products and multiply warped products with a quarter-symmetric connection with constant scalar curvature. Then apply our results to generalized Robertson-Walker spacetimes with a q…

New invariant measures how many twists are needed to unknot welded knots.

Given a virtual link diagram $D$, we define its unknotting index $U(D)$ to be minimum among $(m, n)$ tuples, where $m$ stands for the number of crossings virtualized and $n$ stands for the number of classical crossing changes, to obtain a trivial link diagram. By using span of a diagram and linking number of a diagram …

In this paper we study the warped product submanifolds of a Lorentzian paracosymplectic manifold and obtain some nonexistence results. We show that a warped product semi-invariant submanifold in the form {$M=M_{T}\times_{f}M_{\bot}$} of Lorentzian paracosymplectic manifold such that the characteristic vector field is n…

Estimates heights of special surfaces in warped products.

In this paper, we introduce horizontal and vertical warped product Finsler manifold. We prove that every C-reducible or proper Berwaldian doubly warped product Finsler manifold is Riemannian. Then, we find the relation between Riemmanian curvatures of doubly warped product Finsler manifold and its components, and consi…

Paper defines new submanifolds in Kaehler manifolds.