The Singular Temperley-Lieb Category

We introduce and study the singular Temperley-Lieb category over ℤ[q,q-1], which is a free pivotal category over two self-dual generators and is an extension of the (classical) Temperley-Lieb category. Our construction is motivated by a state model for the sl(2) polynomial of an oriented link and provides a categorical perspective to this link invariant. We also construct a couple of polynomial invariants for oriented tangles from category theory point of view.

Postulation of General Unions of Lines and Multiplicity Two Points in ℙ3

We prove that a general union X⊂ℙ3 of prescribed numbers of lines and double points has maximal rank, except a few well-known exceptional cases.

Totally Umbilical Lightlike Hypersurfaces in Robertson-Walker Spacetimes

We study the problem of lightlike hypersurface immersed into Robertson-Walker (RW) spacetimes in this paper, where the screen bundle of the hypersurface has constant higher order mean curvature. We consider the following question: under what conditions is the compact lightlike hypersurface totally umbilical? Our approach is based on the relationship between the lightlike hypersurface with its screen bundle and the Minkowski formulae for the screen bundle.

On the Class of (q,α,β)-Metrics

We study a class of Finsler metrics in the form F=α+βq/αq-1, where α is a Riemannian metric, β is a 1-form, and 1<q<2. F is called (q,α,β)-metrics. We find the necessary and sufficient conditions under which the class of (q,α,β)-metrics is locally projectively flat and Douglas metrics, respectively.

A Survey on Seifert Fiber Space Theorem

We review the history of the proof of the Seifert fiber space theorem, as well as its motivations in 3-manifold topology and its generalizations.

Submanifolds of Sasakian Manifolds with Certain Parallel Operators

We study submanifolds of Sasakian manifolds and obtain a condition under which certain naturally defined symmetric tensor field on the submanifold is to be parallel and use this result to obtain conditions under which a submanifold of the Sasakian manifold is an invariant submanifold.

Rank 1 Decompositions of Symmetric Tensors Outside a Fixed Support

Let νd:ℙm→ℙn, n:=(n+dn)-1, denote the degree d Veronese embedding of ℙm. For any P∈ℙn, let sr(P) be the minimal cardinality of S⊂νd(ℙm) such that P∈〈S〉. Identifying P with a homogeneous polynomial q (or a symmetric tensor), S corresponds to writing q as a sum of ♯(S) powers Ld with L a linear form (or as a sum of ♯(S) d-powers of vectors). Here we fix an integral variety T⊊ℙm and P∈〈νd(T)〉 and study a similar decomposition with S⊈T and ♯(S) minimal. For instance, if T is a linear subspace, then we prove that ♯(S)≥♯(S∩T)+d+1 and classify all (S,P) such that ♯(S)-♯(S∩T)≤2d-1.

Da-Homothetic Deformation of K-Contact Manifolds

We study Da-homothetic deformations of K-contact manifolds. We prove that Da-homothetically deformed K-contact manifold is a generalized Sasakian space form if it is conharmonically flat. Further, we find expressions for scalar curvature of Da-homothetically deformed K-contact manifolds.

Postulation of a Union X⊂ℙr, r≥4, of a Given Zero-Dimensional Scheme and Several General Lines

We study the Hilbert function of the union X of a given zero-dimensional scheme Z⊂ℙr, r≥4, and t general lines of ℙr, proving, for example, that if t≫deg(Z), then X has maximal rank.