Space times of embedding class one in general relativity

The algebraic structures of both the Ricci and Weyl tensors, given that of the second fundamental tensor, are tabulated. In particular, the Weyl tensor is algebraically special if and only if the second fundamental tensor is algebraically special. A class one perfect fluid is found to possess at least one of the following properties: (a) conformai flatness; (b) the flow is geodesic; (c) it admits a three-dimensional group of isometries with two-dimensional space-like trajectories. All solutions with property (b) are obtained explicitly.