Ricci curvature for conjugate and focal points on GRW space-times

摘要

The authors compute the Ricci curvature of the GRW space-time to obtain two conditions for the conjugate points which appear as the Timelike Convergence Condition(TCG) and the Jacobi inequality. Moreover, under such two conditions, we obtain a lower bound of the length of a unit timelike geodesic for focal points emanating from the immersed spacelike hypersurface, the graph over the fiber in the GRW space-time.