Given a smooth function and a Riemannian metric on a compact andboundaryless manifold, one can define a gradient vector field. Undergeneric assumptions (of Morse-Smale type), I will explain how one cancompute explicitely the spectrum of the associated Lie derivativeacting on certain anisotropic Sobolev spaces of currents. I will alsodiscuss the relation of this spectral analysis with differentialtopology. This is a joint work with Nguyen Viet Dang (University Lyon1).