On the Ricci curvature tensor of (k+1)- dimensional semi-ruled surfaces in $E^{n+1}_v$

Abstract

If we choose a natural companion basis for (k +1) -dimensional semi-ruled surfaces in semi-Euclidean space $E^{n+1}_v$, then the metric coefficients are $g_{ij}= \varepsilon_i \delta_{ij}, 1\leq i,j \leq k$. In this paper we show that the Ricci curvature tensor of a (k + 1) -dimensional semi-ruled surface in the semi-Euclidean space $E^{n+1}_v$ is $S=\sum_{j,h=0}^k\varepsilon_{jh}(\varepsilon_0R_{h0}^0 {_{j}g_{00}} + \sum_{i=0}^kR_{hij}^i$ + $\sum_{i=0}^kg_{i0}(\varepsilon_iR_{hij}^0 + \varepsilon_0R_{h0j}^i))\theta_j \bigotimes \theta_h$ Here, $\{\theta_0,\theta_1,...,\theta_k\}$ is the dual basis of the local coordinate basis $\{e_0,e_1,...,e_k\}$.