The integration of the equations for geodesics of left-invariant metrics on simple Lie groups using special functions

This paper studies a multiparameter family of left-invariant metrics on simple Lie groups which generalizes the inertia tensor of an $n$-dimensional rigid body. A class of solutions is produced for the geodesic equations on simple linear groups expressed in terms of quasipolynomials. For groups of complex matrices with determinant one, explicit formulas are found for the matrix elements of geodesics. The matrix elements are polynomials in exponentials and in theta-functions on Riemann surfaces.
Bibliography: 11 title