The Initial value problem using metric and extrinsic curvature

The initial value problem is introduced after a thorough review of the essential geometry. The initial value equations are put into elliptic form using both conformal transformations and a treatment of the extrinsic curvature introduced recently. This use of the metric and the extrinsic curvature is manifestly equivalent to the author's conformal thin sandwich formulation. Therefore, the reformulation of the constraints as an elliptic system by use of conformal techniques is complete.