Extension of Riemannian metrics and growth type (Q1284728)

Let \(M\) be a \((p+m)\)-dimensional noncompact manifold and \(V\) a \(p\)-dimensional submanifold. Assume that \(V\) is equipped with a complete Riemannian metric \(g\). The normal bundle \(E= TM/TV\) of \(V\) can be realized as a closed tubular neighbourhood of \(V\). Let \(\{N_n\}\) be a family of compact submanifolds which constitutes an exhaustion of \(M-E^\circ\) (the interior of \(E\)). The authors first construct a metric \(g_E\) on \(E\), such that \(g_E\) has the same growth-type as \(g\). Secondly, they choose a metric \(\sigma_n\) on each \(N_n\) to keep the diameter and volume of \(N_n\) finite. Finally, they choose a metric \(g_{T_n}\) on each compact intermediate \(T_n\) between \(P_n\) \((=N_n\times [-1,1])\) and \(P_{n-1}\), to assure that the remainders of diameters and of volumes of \(T_n\) tend to zero rapidly. In this way, the authors obtain a complete Riemannian metric \(G\) on \(M\), having the same growth-type as \(g\).