• entropy
• riemannian manifold
• symmetric space
• rigidity
• mostow

This thesis is based on the article "Entropie minimale et rigiditès des espaces localement symétriques de courbure strictement négative" of G.Besson, G.Courtois and S.Gallot. We consider a compact oriented n-Rimennian manifold M that supports a locally symmetric metric of negative sectional curvature. The aim of the thesis is to show that an invariant characterizes this metric.
The invariant that we consider is the volume entropy of a metric g on a compact Riemannian manifold. It measures the growth of the volume of the balls B(x,r) of the covering manifold of M endowed with the cover metric when r tends to infinity. The entropy of a metric g is denoted by h(g).
The quantity ent(g)=h(g)^n*Vol(M,g), where Vol(M,g) is the volume of M calculated with respect to g, is invariant under homotheties.
The main result of the thesis is that the locally symmetric metric of negative curvature on M minimizes the quantity ent(g), where g varies among all the Riemannian metrics on M. Moreover it is unique in the following sense: if g is a Riemannian metric that minimizes the functional ent, then, up to homotheties, the metric g is isometric to the locally symmetric one.
This result give us some rigidity theorems on Riemannian manifolds that support a locally symmetric metric of negative curvature. For example a corollary of the theorem above is the well known Mostow rigidity theorem.