Some Characterization of Spiral Surfaces

In this paper we will study special spiral surfaces in the three dimensional Euclidean space and we give some characterization of these surfaces. More specifically, we investigate the Chang-Yau operator acting on the Gauss map of spiral surfaces. We also give some results about canonical vector field of these surfaces, i.e., we study incomperssibility of canonical vector field in two types of spiral surfaces. Moreover, we give some necessary conditions for a spiral surface to be a Weingarten surface. Existence of umbilical point is another problem that we investigate about it for a special case of spiral surfaces of the first type.

A link between harmonicity of 2-distance functions and incompressibility of canonical vector fields

Let $M$ be a Riemannian submanifold of a Riemannian manifold $\tilde M$ equipped with a concurrent vector field $\tilde Z$. Let $Z$ denote the restriction of $\tilde Z$ along $M$ and let $Z^T$ be the tangential component of $Z$ on $M$, called the canonical vector field of $M$. The 2-distance function $\delta^2_Z$ of $M$ (associated with $Z$) is defined by $\delta^2_Z=\$. In this article, we initiate the study of submanifolds $M$ of $\tilde M$ with incompressible canonical vector field $Z^T$ arisen from a concurrent vector field $\tilde Z$ on the ambient space $\tilde M$. First, we derive some necessary and sufficient conditions for such canonical vector fields to be incompressible. In particular, we prove that the 2-distance function $\delta^2_Z$ is harmonic if and only if the canonical vector field $Z^T$ on $M$ is an incompressible vector field. Then we provide some applications of our main results.

Order of the canonical vector bundle over configuration spaces of spheres

Given a vector bundle, its (stable) order is the smallest positive integer t such that the t-fold self-Whitney sum is (stably) trivial. So far, the order and the stable order of the canonical vector bundle over configuration spaces of Euclidean spaces have been studied in [F. R. Cohen, R. L. Cohen, N. J. Kuhn and J. A. Neisendorfer, Bundles over configuration spaces, Pacific J. Math. 104 1983, 1, 47–54], [F. R. Cohen, M. E. Mahowald and R. J. Milgram, The stable decomposition for the double loop space of a sphere, Algebraic and Geometric Topology (Stanford 1976), Proc. Sympos. Pure Math. 32 Part 2, American Mathematical Society, Providence 1978, 225–228], and [S.-W. Yang, Order of the Canonical Vector Bundle on {C_{n}(k)/\Sigma_{k}}, ProQuest LLC, Ann Arbor, 1978]. Moreover, the order and the stable order of the canonical vector bundle over configuration spaces of closed orientable Riemann surfaces with genus greater than or equal to one have been studied in [F. R. Cohen, R. L. Cohen, N. J. Kuhn and J. A. Neisendorfer, Bundles over configuration spaces, Pacific J. Math. 104 1983, 1, 47–54]. In this paper, we mainly study the order and the stable order of the canonical vector bundle over configuration spaces of spheres and disjoint unions of spheres.

Dual-depth Adapted Irreducible Formal Multizeta Values

Let $\mathfrak{ds}$ denote the double shuffle Lie algebra, equipped with the standard weight grading and depth filtration; we write $\mathfrak{ds}=\oplus_{n\ge 3} \mathfrak{ds}_n$ and denote the filtration by $\mathfrak{ds}^1\supset \mathfrak{ds}^2\supset \cdots$. The double shuffle Lie algebra is dual to the new formal multizeta space $\mathfrak{nfz}=\oplus_{n\ge 3} \mathfrak{nfz}_n$, which is equipped with the dual depth filtration $\mathfrak{nfz}^1\subset \mathfrak{nfz}^2\subset\cdots$ Via an explicit canonical isomorphism $\mathfrak{ds}\buildrel \sim\over\rightarrow\mathfrak{nfz}$, we define the "dual" in $\mathfrak{nfz}$ of an element in $\mathfrak{ds}$. For each weight $n\ge 3$ and depth $d\ge 1$, we then define the vector subspace $\mathfrak{ds}_{n,d}$ of $\mathfrak{ds}$ as the space of elements in $\mathfrak{ds}_n^d-\mathfrak{ds}_n^{d+1}$ whose duals lie in $\mathfrak{nfz}_n^d$. We prove the direct sum decomposition \[ \mathfrak{ds}=\bigoplus_{n\ge 3}\bigoplus_{d\ge 1} \mathfrak{ds}_{n,d}, \] \noindent which yields a canonical vector space isomorphism between $\mathfrak{ds}$ and its associated graded for the depth filtration, $\mathfrak{ds}_{n,d}\simeq \mathfrak{ds}_n^d/ \mathfrak{ds}_n^{d+1}$. A basis of $\mathfrak{ds}$ respecting this decomposition is dual-depth adapted, which means that it is adapted to the depth filtration on $\mathfrak{ds}$, and the basis of dual elements is adapted to the dual depth filtration on $\mathfrak{nfz}$. We use this notion to give a canonical depth 1 generator $f_n$ for $\mathfrak{ds}$ in each odd weight $n\ge 3$, namely the dual of the new formal single zeta value $\zeta(n)\in\mathfrak{nfz}_n$. At the end, we also apply the result to give canonical irreducibles for the formal multizeta algebra in weights up to 12.

The Geometry of Tangent Bundles: Canonical Vector Fields

A canonical vector field on the tangent bundle is a vector field defined by an invariant coordinate construction. In this paper, a complete classification of canonical vector fields on tangent bundles, depending on vector fields defined on their bases, is obtained. It is shown that every canonical vector field is a linear combination with constant coefficients of three vector fields: the variational vector field (canonical lift), the Liouville vector field, and the vertical lift of a vector field on the base of the tangent bundle.