  In the context of computer code experiments, sensitivity analysis of a
complicated input-output system is often performed by ranking the so-called
Sobol indices. One reason of the popularity of Sobol's approach relies on the
simplicity of the statistical estimation of these indices using the so-called
Pick and Freeze method. In this work we propose and study sensitivity indices
for the case where the output lies on a Riemannian manifold. These indices are
based on a Cram\'er von Mises like criterion that takes into account the
geometry of the output support. We propose a Pick-Freeze like estimator of
these indices based on an $U$--statistic. The asymptotic properties of these
estimators are studied. Further, we provide and discuss some interesting
numerical examples.
