Geometry of hyperspace. I

Abstract

Hyperspace is heuristically defined as an (infinitely dimensional) manifold of all spacelike hypersurfaces embedded in a given Riemannian spacetime. The Riemannian structure (M,g) of spacetime induces a rich geometrical structure in hyperspace. Part of that structure, especially the moving normal frames in hyperspace, Lie derivatives, and symmetrical ∇ and asymmetrical ∇* covariant hyperderivatives, are studied in detail. The formalism introduced in this paper sets the stage for the geometrical study of hypersurface kinematics and dynamics of general tensor fields with derivative gravitational coupling, and of the Dirac–ADM geometrodynamics with such tensor sources, in the following papers.