Appears in collections : Geometry and dynamics of Finsler manifolds / Géométrie et dynamiques des espaces de Finsler, Geometry and dynamics of Finsler manifolds / Géométrie et dynamiques des espaces de Finsler

I will discuss some projective differential geometric invariants of properly convex domains arising from affine dfferential geometry. Consider a properly convex domain $\Omega $ in $R^n\subset RP^n$, and the cone $C$ over $\Omega $ in $R^{n+1}$. Then Cheng-Yau have shown that there is a unique hyperbolic affine sphere which is contained in $C$ and asymptotic to the boundary $\partial C$. The hyperbolic affine sphere is invariant under special linear automorphisms of $C$ , and carries an invariant complete Riemannian metric of negative Ricci curvature, the Blaschke metric. The Blaschke metric descends to a projective-invariantmetric on $\Omega $. I will also address the relationship between the Blaschke metric and Hilbert metric, which is recent and is due to Benoist-Hulin. At the end, I will discuss applications to the geometry of real projective structures on surfaces.