  We glue two manifolds which have curvature operators at least k (in the sense
of eigenvalues) along their common boundary. We show that if the sum of the
second fundamental forms of the boundary is positive semidefinite, then the
curvature operator of the resulting manifold is at least k up to an arbitrarily
small error term. Similar results hold for Ricci, scalar, bi, isotropic and
flag curvature, respectively.
