SUPERSYMMETRIC QUANTUM-THEORY AND DIFFERENTIAL GEOMETRY

We reconsider differential geometry from the point of view of the quan tum theory of non-relativistic spinning particles, which provides exam ples of supersymmetric quantum mechanics. This enables us to encode ge ometrical structure in algebraic data consisting of an algebra of func tions on a manifold and a family of supersymmetry generators represent ed on a Hilbert space. We show that known types of differential geomet ry can be classified in terms of the supersymmetries they exhibit. Our formulation is tailor-made for a generalization to non-commutative ge ometry, which will be presented in a separate paper.