Intrinsic characterization of certain conformally flat spaces

Abstract

If Vn is a conformally flat hypersurface of a conformally flat space Vn+i, then Vn is a quasi-umbilical hypersurface, that is, there exists a non-zero vector field Vi on Vn such that the second fundamental tensor hμ is given by hμ=agμ+βvjVi for some functions a, β on Fn, here gμ is the metric tensor on Vn (See [1]). If Vn^i is a space of constant curvature k, Chen and Yano showed in [1] that the curvature tensor Kkμ h is given by