  Using results on inverse spectral problems, in particular the so-called new
wave invariants attached to a classical equilibrium, we show that it is
possible to determine the Morse index of height functions. For compact
Riemannian surfaces $M\subset \mathbb{R}^3$ this imply that we can retrieve the
topology (via the genus).
  Our results are independent from the choice of a metric on $M$ and can be
obtained from the choice of a 'generic' height-function. For surfaces of genus
zero, diffeomorphic to a 2-sphere, the method allows to detect the convexity,
or the local convexity of the surface.
  Keywords : Micro-local analysis; Schr\"odinger operators; Inverse spectral
problems.
