Indices of locally minimal networks on a sphere (Q1866856)

In this paper the classical notions of Morse index and zero-index of geodesic networks [\textit{J. Milnor}, English orig. Princeton Univ. Press (1963; Zbl 0108.10401); \textit{D. Gromoll, W. Klingenberg} and \textit{W. Meyer}, ``Riemannsche Geometrie im Großen''. Springer-Verlag, Berlin (1968; Zbl 0155.30701)] are generalized for the case of locally minimal networks. The main result of the paper is as follows. Let \(\Gamma\) be a closed locally minimal network on \(S^2\), different from a closed geodesic. Then the Morse index of the network \(\Gamma\) is equal to the cyclomatic number (the first Betti number) of the network and the zero-index is equal to three.