  We give a definition of the fractional Laplacian on some noncompact
manifolds, through an extension problem introduced by Caffarelli-Silvestre.
While this definition in the compact case is straightforward, in the noncompact
setting one needs to have a precise control of the behavior of the metric at
infinity and geometry plays a crucial role. First we give explicit calculations
in the hyperbolic space, including a formula for the kernel and a trace Sobolev
inequality. Then we consider more general noncompact manifolds, where the
problem reduces to obtain suitable upper bounds for the heat kernel.
