Let (M, g) be a simple Riemannian manifold with boundary and consider the geodesic ray transform of symmetric 2-tensor fields. Let the integral of f along maximal geodesics vanish on an appropriate open subset of the space of geodesics in M . Under the assumption that the metric g is real-analytic, it is shown that there exists a vector field v satisfying f = dv on the set of points lying on th...

The computation of the index of the Hessian of the action functional in semi-Riemannian geometry at geodesics with two variable endpoints is reduced to the case of a fixed final endpoint. Using this observation, we give an elementary proof of the Morse Index Theorem for Riemannian geodesics with two variable endpoints, in the spirit of the original Morse’s proof. This approach reduces substanti...

This paper discusses the extent to which one can determine the space-time metric from a knowledge of a certain subset of the (unparametrised) geodesics of its Levi-Civita connection, that is, from the experimental evidence of the equivalence principle. It is shown that, if the space-time concerned is known to be vacuum, then the Levi-Civita connection is uniquely determined and its associated m...

The space $J^k$ of $k$-jets a real function one variable $x$ admits the structure Carnot group type. As such, submetry (\sR submersion) onto Euclidean plane. Horizontal lifts lines (which are left-translates horizontal one-parameter subgroups) thus globally minimizing geodesics on $J^k$. All $J^k$-geodesics, or not, constructed from degree $k$ polynomials in according to Anzaldo-Meneses and Mon...