Application of topological methods to estimate the number of longitudinal elastic waves in crystals

In the problem of estimating the number of longitudinal normals of elastic waves in crystals, several approaches based on the methods of modern topology and associated with the use of topological invariants are proposed. It is shown that for any symmetry class of the crystals there exists not less than three directions along which purely longitudinal waves can propagate. Under the Morse condition on the fundamental function, connections are obtained between the numbers of directions of longitudinal normals corresponding to different types of critical points of this function. Further prospects are discussed.