On the Ricci Curvature Tensor of (k+1)-Dimensional Semi Ruled Surfaces Evn+1

If we choose a natural companion basis for \((k+1)\)-dimensional semi-ruled surfaces in semi-Euclidean space \(E_{\nu}^{n+1}\), then the metric coefficients are \(g_{ij} = \epsilon_{i}\delta_{ij}\), \(1\leq i\), \(j \leq k\). In this paper we show that the Ricci curvature tensor of a \((k+1)\)-dimensional semi-ruled surfaces in semi-Euclidean space \(E_{\nu}^{n+1}\) is \( S = \sum_{j,h=0}^{k}{\epsilon_{jh}} \left( \epsilon_{0} R_{h-j}^{0} g_{00} + \sum_{i=0}^{k}{R_{hij}^{i}} + \sum_{i=0}^{k}{g_{i0}} \left(\epsilon_{i} R_{hij} + \epsilon_{0} R_{h0j}^{i} \right)\right) \theta_{j} \otimes \theta_{h}. \) Here, \(\{\theta_{0}, \theta_{1}, ... \theta_{k}\}\) is the dual basis of the local coordinate basis \(\{e_{0} , e_{1} , ...e_{k}\}\).