(R1519) On Some Geometric Properties of Non-null Curves via its Position Vectors in \mathbb{R}_1^3

In this work, the geometric properties of non-null curves lying completely on spacelike surface via its position vectors in the dimensional Minkowski 3-space \mathbb{R}_1^3 are studied. Also, we give a few portrayals for the spacelike curves which lie on certain subspaces of \mathbb{R}_1^3. Finally, we present an application to demonstrate our insights

On geometry of spherical image in Minkowski space-time with timelike type-2 parallel transport frame

In this paper, we investigate a different type of the parallel transport frame in 3-dimensional Minkowski space R^3_1 by using the binormal vector field of a timelike regular curve as common vector field to introduce, and we recall this frame as timelike type-2 parallel transport frame . Also, we present new spherical images and call them as timelike type-2 parallel transport spherical images by translating the induced frame vectors to the center of unit Lorentzian sphere in 3-dimensional Minkowski space R^3_1. Additionally, we obtain the Frenet apparatus of these new spherical images in terms of base curves timelike type-2 parallel transport invariants. Finally, interesting relations are expressed and illustrate an example of the results

Characterizing non-totally geodesic spheres in a unit sphere

A concircular vector field $\mathbf{u}$ on the unit sphere $\mathbf{S}^{n+1}$ induces a vector field $\mathbf{w}$ on an orientable hypersurface $M$ of the unit sphere $\mathbf{S}^{n+1}$, simply called the induced vector field on the hypersurface $M$. Moreover, there are two smooth functions, $f$ and $\sigma$, defined on the hypersurface $M$, where $f$ is the restriction of the potential function $\overline{f}$ of the concircural vector field $\mathbf{u}$ on the unit sphere $\mathbf{S}^{n+1}$ to $M$ and $\sigma$ is defined as $g\left(\mathbf{u}, N\right)$, where $N$ is the unit normal to the hypersurface. In this paper, we show that if function $f$ on the compact hypersurface satisfies the Fischer–Marsden equation and the integral of the squared length of the vector field $\mathbf{w}$ has a certain lower bound, then a characterization of a small sphere in the unit sphere $\mathbf{S}^{n+1}$ is produced. Additionally, we find another characterization of a small sphere using a lower bound on the integral of the Ricci curvature of the compact hypersurface $M$ in the direction of the vector field $\mathbf{w}$ with a non-zero function $\sigma$

The 2016 Season of the Al-Wajh – Al-'Ula Survey Project : Preliminary Report

Peer reviewe

Geometry of Tangent Poisson–Lie Groups

Let G be a Poisson–Lie group equipped with a left invariant contravariant pseudo-Riemannian metric. There are many ways to lift the Poisson structure on G to the tangent bundle TG of G. In this paper, we induce a left invariant contravariant pseudo-Riemannian metric on the tangent bundle TG, and we express in different cases the contravariant Levi-Civita connection and curvature of TG in terms of the contravariant Levi-Civita connection and the curvature of G. We prove that the space of differential forms Ω*(G) on G is a differential graded Poisson algebra if, and only if, Ω*(TG) is a differential graded Poisson algebra. Moreover, we show that G is a pseudo-Riemannian Poisson–Lie group if, and only if, the Sanchez de Alvarez tangent Poisson–Lie group TG is also a pseudo-Riemannian Poisson–Lie group. Finally, some examples of pseudo-Riemannian tangent Poisson–Lie groups are given

Estimation of Ricci Curvature for Hemi-Slant Warped Product Submanifolds of Generalized Complex Space Forms and Their Applications

In this paper, we estimate Ricci curvature inequalities for a hemi-slant warped product submanifold immersed isometrically in a generalized complex space form with a nearly Kaehler structure, and the equality cases are also discussed. Moreover, we also gave the equivalent version of these inequalities. In a later study, we will exhibit the application of differential equations to the acquired results. In fact, we prove that the base manifold is isometric to Euclidean space under a specific condition

The Heat Equation on Submanifolds in Lie Groups and Random Motions on Spheres

We studied the random variable Vt=volS2(gtB∩B), where B is a disc on the sphere S2 centered at the north pole and (gt)t≥0 is the Brownian motion on the special orthogonal group SO(3) starting at the identity. We applied the results of the theory of compact Lie groups to evaluate the expectation of Vt for 0≤t≤τ, where τ is the first time when Vt vanishes. We obtained an integral formula using the heat equation on some Riemannian submanifold ΓB seen as the support of the function f(g)=volS2(gB∩B) immersed in SO(3). The integral formula depends on the mean curvature of ΓB and the diameter of B

A Note on Minimal Hypersurfaces of an Odd Dimensional Sphere

We obtain the Wang-type integral inequalities for compact minimal hypersurfaces in the unit sphere S 2 n + 1 with Sasakian structure and use these inequalities to find two characterizations of minimal Clifford hypersurfaces in the unit sphere S 2 n + 1

Characterizing spheres by an immersion in Euclidean spaces

In this paper we study compact immersed orientable hypersurfaces in the Euclidean space Rn+1 and show that suitable restrictions on the tangential and normal components of the immersion give different characterizations of the spheres