Some new examples of nonlinear quotients with nonnegative sectional curvature

In this note, new examples of manifolds with Riemannian metrics of nonnegative sectional curvature are constructed. These manifolds are orbit spaces of free nonlinear orientation-preserving and orientation-reversing involutions on products of spheres, their fundamental group is of order two, and they realize homeomorphism classes that were not previously known to contain nonnegatively curved smooth manifolds. We remark that our examples are obtained as a cut-and-paste construction along spheres and real projective spaces on sphere bundles.