On the Curvatures of (k+1)-Dimensional Semi-Ruled Surfaces in \({{E}_{v}^{n+1} }\)

Abstract

In this paper, first we will define the generalized (k + 1) -dimensional semi-ruled surface M*, such that the generator space of M* is the semi-subspace of E v n + 1 where E v n + 1 is the semi-Euclidean space. Then, we will compute the mean curvature, Riemann curvature, Ricci curvature and scalar curvature of M*.

Beltrami-Euler Formulas of Generalized Semi-Ruled Surface in Semi Euclidean Space

“…It is clear that A (t) is a semi-subspace. Also, for the orthonormal base {e 1 (t) , e 2 (t) , ..., e k (t)} there are the following equations, [4],…”

“…The tangential bundle T (t) has at least number of µ timelike vectors. Therefore, T (t) is a semi-subspace, [4].…”

“…They obtained the geodesic curvature, the geodesic torsion of a curve and the normal curvature. The (k + 1) −dimensional generalized semi ruled surfaces in E n+1 v were given by [4]. In this work, we have studied the sectional curvatures of the generalized semi ruled surfaces in (n + 1) −dimensional semi Euclidean space E n+1 v .…”

“…Here, the subspace E k,µ (t) and the curve α are called the generating space and base curve, respectively, [4]. Throughout this paper, we assume that E k,µ (t), µ ≥ 1, is a semi-subspace and…”

“…, e 1 (t) , ..., e k (t) is linearly independent and the base curve α is a non-null curve, [4].…”