On some pasting cylinders onto a manifold with negative (Ricci, scalar) curvature along compact boundaries (Q2639352)

The author shows that if M is a negatively Ricci (resp. scalar) curved complete manifold with compact boundary \(\partial M\), and \(\tilde M\) is a manifold without boundary obtained by pasting cylinders onto M, then there is a complete metric g on \(\tilde M\) with negative Ricci (resp. scalar) curvature such that the inclusion \(M\subset \tilde M\) is an isometric imbedding and g is a warped product metric on \(\tilde M\setminus S\) (S is an open neighborhood of M) whose vertical fibres are homothetic to \(\partial M\).