Riemannian geometry in spaces with Grassman coordinates

The use of spaces containing Grassman (anticommuting) coordinates (in addition to the usual space-time coordinates) as a framework for unified gauge theories is described. The theory developed represents a local gauge-invariant extension of conventional (global) supersymmetry. Aside from containing the usual general coordinate invariance group of gravitational theory, the gauge supersymmetry group is seen to also encompass other symmetries of particle physics, e.g., electromagnetic (or Yang-Mills) invariance. The role of spontaneous symmetry breaking and the field equations unifying the Einstein, Maxwell, and Dirac interactions are discussed.Research supported in part by the National Science Foundation.Invited talk at the conference, The Riddle of Gravitation, on the Occasion of the 60th Birthday of Peter G. Bergmann, Syracuse, New York, March 1975.