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In many applied problems one seeks to identify and count the critical points of a particular eigenvalue of a smooth parametric family of self-adjoint matrices, with the parameter space often being known and simple, such as a torus. Among particular settings where such a question arises are the Floquet–Bloch decomposition of periodic Schrödinger operators, topology of potential energy surfaces in quantum chemistry, spectral optimization problems such as minimal spectral partitions of manifolds, as well as nodal statistics of graph eigenfunctions. In contrast to the classical Morse theory dealing with smooth functions, the eigenvalues of families of self-adjoint matrices are not smooth at the points corresponding to repeated eigenvalues (called, depending on the application and on the dimension of the parameter space, the diabolical/Dirac/Weyl points or the conical intersections). This work develops a procedure for associating a Morse polynomial to a point of eigenvalue multiplicity; it utilizes the assumptions of smoothness and self-adjointness of the family to provide concrete answers. In particular, we define the notions of non-degenerate topologically critical point and generalized Morse family, establish that generalized Morse families are generic in an appropriate sense, establish a differential first-order conditions for criticality, as well as compute the local contribution of a topologically critical point to the Morse polynomial. Remarkably, the non-smooth contribution to the Morse polynomial turns out to depend only on the size of the eigenvalue multiplicity and the relative position of the eigenvalue of interest and not on the particulars of the operator family; it is expressed in terms of the homologies of Grassmannians. 

The classical result of Eisenhart states that, if a Riemannian metric admits a Riemannian metric that is not constantly proportional to and has the same (parameterized) geodesics as in a neighborhood of a given point, then is a direct product of two Riemannian metrics in this neighborhood. We introduce a new generic class of step graded nilpotent Lie algebras, called -surjective, and extend the Eisenhart theorem to sub-Riemannian metrics on step distributions with -surjective Tanaka symbols. The class of ad-surjective step nilpotent Lie algebras contains a well-known class of algebras of H-type as a very particular case. 

We prove that for every n ≥ 3 the sharp upper bound for the dimension of the symmetry groups of homogeneous, 2-nondegenerate, (2n+1)-dimensional CR manifolds of hypersurface type with a 1-dimensional Levi kernel is equal to n^2+7, and simultaneously establish the same result for a more general class of structures characterized by weakening the homogeneity condition. This supports Beloshapka’s conjecture stating that hypersurface models with a maximal finite-dimensional group of symmetries for a given dimension of the underlying manifold are Levi nondegenerate.