Parabolic PDE in Curved Spaces: Exploring Solutions on Manifolds

[gvk]

The parabolic approximation was introduced by Leontovich and Fock in 1946 to describe the propagation of the electromagnetic waves in the Earth atmosphera (see Levy M. Parabolic equation methods for electromagnetic wave propagation, 2000). However, the parabolic equation was known long before that, e.g. the time dependent Schr¨odinger equation in QM is the parabolic PDE.
In both cases the parabolic equation are considered in orthogonal Cartesian coordinate system.
Does anybody know the consideration of the parabolic equation in the space with the curvature, viz. in curvilinear coordinate system? The book by Levy does not have any references in PDE.
Thank you for any hint.

 

[Reb]

Does anybody know the consideration of the parabolic equation in the space with the curvature, viz. in curvilinear coordinate system?

On a manifold, that is...
I know http://www.cambridge.org/US/catalogue/catalogue.asp?isbn=9780521409971" book that deal with qualitative properties of the solution of the heat equation on a manifold (which, since the Laplacian depends on the metric, becomes non-constant coefficient when expressed in local coordinates). For more general considerations, the topology of the manifold comes naturally into play.