The restriction theorem for fully nonlinear subequations

Let X be a submanifold of a manifold Z. We address the question: When do viscosity subsolutions of a fully nonlinear PDE on Z, restrict to be viscosity subsolutions of the restricted subequation on X? This is not always true, and conditions are required. We first prove a basic result which, in theory, can be applied to any subequation. Then two definitive results are obtained. The first applies to any "geometrically defined" subequation, and the second to any subequation which can be transformed to a constant coefficient (i.e., euclidean) model. This provides a long list of geometrically and analytically interesting cases where restriction holds.