Hypergravitational field equations

We extend Einstein’s field equations and Hilbert’s action for gravity to the hyperspace of dimension N² that is built out of N-component spinors in the way that four-dimensional time-space is built out of two-component ones. The hypergravitational field equation is of differential order N in a chronometric tensor with N indices. The natural spin structures of the unitary groups U(N) define Einstein-type cosmologies, static, U(N)-invariant, and closed, that satisfy the extended equations with neither dust nor cosmological constant, except for N=2 which requires both.