ξ a parameter representing the choice of direction of the geodesic in Sn−1. Then the spherical Laplacian has the form: where is the Laplace–Beltrami operator on the ordinary unit -sphere. In particular, for the ordinary 2-sphere using standard notation for polar coordinates we get: Hyperbolic space A similar technique works in hyperbolic space. Here the hyperbolic space Hn−1 can be embedded into the n dimensional Minkowski space, a real vector space equipped with the quadratic form Then Hn is the subset of the future null cone in Minkowski space given by Then Here is the degree zero homogeneous extension of f to the interior of the future null cone and is the wave operator The operator can also be written in polar coordinates. Let be spherical coordinates on the sphere with respect to a particular point p of Hn−1 (say, the center of the Poincaré disc). Here t represents the hyperbolic distance from p and ξ a parameter representing the choice of direction of the geodesic in Sn−2. Then the hyperbolic Laplacian has the form: where is the Laplace–Beltrami operator on the ordinary unit (n − 2)-sphere. In particular, for the hyperbolic plane using standard notation for polar coordinates we get: See also Covariant derivative Laplacian operators in differential geometry Laplace operator Notes References . Differential operators Riemannian geometry de:Verallgemeinerter Laplace-Operator#Laplace-Beltrami-Operator