LOCAL AND INFINITESIMAL RIGIDITY OF SIMPLY CONNECTED NEGATIVELY CURVED MANIFOLDS

Let (X, g(0)) be a simply connected Riemannian manifold with sectional curvature K <= -1. For a metric g on X which is equal to go outside a compact the identity map of X induces a conformal map (id) over cap (g0,g) : partial derivative(g0) X -> partial derivative X-g between the boundaries at infinity of X with respect to go and g. We define a function S(g) on the space of geodesics of (X, g(0)), called the integrated Schwarzian of g, which measures the deviation of this conformal map from being Moebius. We use the integrated Schwarzian to prove local and infinitesimal rigidity results for such metric deformations.