The geometry of the one-dimensional supersymmetric non-linear sigma models

A new class of one-dimensional non-linear sigma models with N supersymmetries is constructed by generalising the action and the supersymmetry transformations of these theories. The study of the geometry of the associated sigma model manifolds reveals new structures which are different from those found in the two-dimensional theories. In particular, certain N=2 models are defined on manifolds with foliations and the existence of an N=3 multiplet is demonstrated. Conventional off-shell non-linear constrained multiplets with actions are given for all these theories.