Calculus on the multivector manifold spin+(p,q)

Abstract

Coordinate-free differential and integral calculus is formulated on the spin groups, regarded as multivector manifolds embedded in Geometric Algebra. The derivative with respect to a point on the manifold is shown to be in the tangent algebra at the inverse point. The curvature of spin(p,q) is calculated and related to the structure constants of its Lie algebra. Volume elements on spin(p,q) are introduced naturally as completely antisymmetrized geometric products of bivectors. They are left invariant like Haar measure and are elements of a multivector-valued Grassman algebra. It is shown that integration in this sense is forbidden for many spin groups. A version of Stokes’s theorem is formulated on spin(p,q).