The present paper is a continuation of the authors' previous paper [\textit{C. Huneke} and \textit{R. Wiegand}, Math. Ann. 299, No. 3, 449-476 (1994; Zbl 0803.13008)]. The authors study rigidity properties of local cohomology and \(\text{Tor}\). Let \(A\) be a local hypersurface domain of dimension \(d\) and \(M\), \(N\) two finitely generated \(A\)-modules. Firstly, the authors show that \(H_{\mathfrak m}^r(M \otimes N) = 0\) for some \(r < d\) implies \(\text{depth } M \otimes N \geq r+1\) -- that is, \(H_{\mathfrak m}^i(M \otimes N) = 0\) for all \(i \leq r\) -- if \(M \otimes N\) satisfies Serre's \((S_{r+1})\)-condition on the punctured spectrum of \(A\), \(M\) and \(N\) have depth at least \(r\) and \(N\) is of finite projective dimension. The rigidity of \(\text{Tor}\) is used to prove this. Secondly the authors apply the theorem above to vector bundles over a regular local ring or on \(\mathbb{P}^n\).