  The two main topics of this text are as follows: Firstly, three modifications
of the theorem of Beltrami will be presented for diffeomorphisms between
Riemannian manifolds and a space form which preserve the geodesic circles, the
geodesic hyperspheres, or the minimal surfaces, respectively. Secondly, it is
defined what it means for an infinitesimal deformation of a metric to preserve
the geodesics up to first order, and a corresponding infinitesimal version of
Beltrami's theorem is given.
