• hyperbolic geometry
• rigidity

The landscape of rigidity problems in the finite-volume case appears clear, and hence one starts to wonder what happens if the complete hyperbolic manifold has infinite volume or even has geodesic boundary. It comes out that studying the rigidity of complete infinite-volume hyperbolic manifolds which are somehow the extension of compact with geodesic boundary ones, is equivalent to studying the rigidity of these last ones. This happen because the extension is such that the fundamental group and the holonomy representation are preserved.
In 2012 Kerckhoff and Storm proved the infinitesimal rigidity, and hence the local rigidity, of compact hyperbolic manifolds with geodesic boundary in dimension n≥4.
The aim of the thesis is to furnish and develop the necessary tools to treat local and infinitesimal rigidity, and then to lead the reader throughout the proof of infinitesimal rigidity of compact hyperbolic manifolds with geodesic boundary in dimension n≥4.
In the first chapter we briefly treat all the preliminary knowledge of hyperbolic and differential geometry needed. In the second chapter we introduce the De Rham cohomology of bundle-valued forms with respect to a flat bundle connection and Hodge theory. The third chapter is devoted to rigidity.