The submanifold \(F^{m+k}\) in the Euclidean space \(E^ n\) is called a k-linear submanifold if its radius vector r can be given in the form: \[ (1)\quad r(u^ 1,u^ 2,...,u^ m,v^ 1,...,v^ k)=\rho (u^ 1,u^ 2,...,u^ m)+\sum_{\alpha}v^{\alpha}s_{\alpha}(u^ 1,u^ 2,...,u^ m), \] where \(\rho\) and \(s_{\alpha}\) are \(C^ 3\)-maps from an m-dimensional domain U into \(E^ n\), \(\alpha =1,...,k\). A k-linear submanifold is called a k-cylinder if the \(s_{\alpha}\) in (1) are constant. The authors find a necessary and sufficient condition for a k- linear submanifold to be a k-cylinder.