  We consider the inverse problem of determining coefficients appearing in
semilinear elliptic equations stated on Riemannian manifolds with boundary
given the knowledge of the associated Dirichlet-to-Neumann map. We begin with a
negative answer to this problem. Owing to this obstruction, we consider a new
formulation of our inverse problem in terms of a rigidity problem. Precisely,
we consider cases where the Dirichlet-to-Neumann map of a semilinear equation
coincides with the one of a linear equation and ask whether this implies that
the equation must indeed be linear. We give a positive answer to this rigidity
problem under some assumptions imposed to the Riemannian manifold and to the
semilinear term under consideration.
