Jan Boman: Local injectivity for generalised Radon transforms

It is known that the weighted plane Radon transform is locally injective if the weight function is real analytic and positive, and that local injectivity fails for another set of weight functions that is dense in the set of all smooth and positive weight functions. The question of local injectivity has also been asked when the set of straight lines is replaced by the set of geodesics with respect to some metric. In 2012 I described a set G of curve families in an open set X in the plane and a set M of weight functions, both invariant with respect to smooth coordinate transformations, and proved that the corresponding Radon transform is locally injective for all curve families from G and all weight functions m from M. This work led to some very interesting geometric questions on curve families in the plane, treated by Elie Cartan and Tresse more than 100 years ago.