Conjugate and cut loci of geodesic flows have a significant interest in Riemannian geometry, but few examples are known especially on manifolds of dimension greater than 2. An interesting example from mechanics is given by the flow of the Euler top, namely, a geodesic flow on SO(3) with a left invariant metric. The (first) conjugate locus can be determined analytically if two of the three moments of inertia of the body are equal. In that case, the conjugate locus is either a segment or circle (if the body is oblate) or a non-injective mapping of an astroid of revolution (if the body is prolate) [Bates and Fasso` 2006, see the link below]. The analytic study of the generic case of distinct moments of inertia is much more difficult, if not even prohibitive. We thus resort to the numerical construction of the conjugate locus, based on accurate numerical integrations of the flow and of its tangent map. The dependency of the conjugate locus on the moments of inertia is studied in a deformation scenario from the symmetric case. (Joint collaboration with L. Bates).
- http://www.math.unipd.it/~fasso/research/papers/CL-1.ps.zip - Article on the symmetric case