• #### waves on a hyperbolic plane

From Roland Franzius@21:1/5 to matmzc%hofstra.edu@gtempaccount.com on Mon Nov 23 15:02:13 2015
<matmzc%hofstra.edu@gtempaccount.com> wrote:

Hi all.  On any smooth surface one can use the metric to define
Christoffel symbols, define a covariant derivative, and a covariant
Laplacian which one can use to set up a wave equation.  For no
particular reason it occurred to me that one might get interesting
solutions of the wave equation on the hyperbolic plane or in hyperbolic 3-space.  A Google search didn't turn up anything.  Does anyone know of
any relevant literature?  I set up the wave equation for the unit disk
in the Poincare and Klein metrics, but in both cases I got a mess that
seems unsolvable (by humble me anyways).  Any thoughts on this?  Or
perhaps my idea that the problem is worth looking into was overly
optimistic?

Its standard.

The geometry is rotional invariant and in polar or spherical
coordinates
the scalar wave equation separates readily producing products of the 2d
or 3d the angular momentum operator times a solution of the radial
function that solves the radial equation
omega^2 psi - f(r)-1 d_r f(r) d_r psi  + l^2/r^2  psi = m^2 psi

where "l^2" is an eigenvalue of the angular momentum operator squared,
in  3D eg

L^2 = -csc tetha d_theta sin tetha  d_theta + m^2/(sin theta)^2

The exact form depends on dimension of course.

--

Roland Franzius

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• From matmzc%hofstra.edu@gtempaccount.com@21:1/5 to All on Sun Nov 22 17:08:59 2015
Hi all.  On any smooth surface one can use the metric to define
Christoffel symbols, define a covariant derivative, and a covariant
Laplacian which one can use to set up a wave equation.  For no
particular reason it occurred to me that one might get interesting
solutions of the wave equation on the hyperbolic plane or in hyperbolic 3-space.  A Google search didn't turn up anything.  Does anyone know of
any relevant literature?  I set up the wave equation for the unit disk
in the Poincare and Klein metrics, but in both cases I got a mess that
seems unsolvable (by humble me anyways).  Any thoughts on this?  Or
perhaps my idea that the problem is worth looking into was overly
optimistic?

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