Variations of graphs in Riemannian manifolds (Q2780950)

Let \(G\) be a graph. Assume that each edge of \(G\) satisfies Hooke's law [see \textit{R. P. Feynman, R. B. Leighton} and \textit{M. Sands}, The Feynman lecture on physics (1965; Zbl 0131.38703)]. Assume also that \(G\) is embedded in a Riemannian manifold \(M\) such that each edge is geodesical. NEWLINENEWLINENEWLINEThe author introduces the notion of a tension vector at each vertex of \(G\) and that of a tension Jacobi field on \(G\). NEWLINENEWLINENEWLINEThe paper deals with the following conjecture: the sum of lengths of tension vector decreases if the graph \(G\) moves along to the tension Jacobi field. If \(M\) has negative curvature and \(G\) is an expanded graph, then the conjecture holds. The author also gives examples of graphs for which the conjecture does not hold.