Geometry of hypersurfaces

Abstract

This chapter is devoted to hypersurfaces, which are at the basis of the 3+1 formalism for general relativity. After introducing the general notion of hypersurface embedded in spacetime, we focus on spacelike hypersurfaces, which are those involved in the 3+1 formalism. We present the first and second fundamental forms, giving rise to the notions of intrinsic and extrinsic curvatures. Finally, we derive the Gauss-Codazzi equations relating the intrinsic and extrinsic curvatures of an hypersurface to the curvature of the ambient spacetime. All results in this chapter are valid for any spacetime endowed with a Lorentzian metric, whether the latter is or not a solution of Einstein equation. © 2012 Springer-Verlag Berlin Heidelberg.