NONNEGATIVE CURVATURE ON PIECEWISE CONSTANT CURVATURE SPACES

We consider piecewise flat and piecewise spherical spaces. We prove that every piecewise flat cobordism which is a product near the boundary and has nonnegative curvature must be trivial in the metric sense. We also obtain several restrictions for piecewise p-spherical manifolds with nonnegative curvature and diameter equal to pip. We prove that such a manifold must be homeomorphic to a sphere and that it is a disjoint union of minimal paths connecting two points, which have length pip.