I investigate certain stationary charged perfect fluid solutions to the Einstein-Maxwell equations in general relativity. The classification of these solutions has led me to the following question:

Does hyperbolic 3-space admit a non-constant harmonic function (i.e., with vanishing Laplacian) that has a gradient of constant norm?

A non-constant affine function (i.e., with vanishing Hessian) would fit this description. However, hyperbolic space does not admit such a function.

T. Sakai: "On Riemannian manifolds admitting a function whose gradient is of constant norm", Kodai Math. J. 19 (1996), 39-51.

I investigate certain stationary charged perfect fluid solutions to the Einstein-Maxwell equations in general relativity. The classification of these solutions has led me to the following question:

Does hyperbolic 3-space admit a non-constant harmonic function (i.e., with vanishing Laplacian) that has a gradient of constant norm?

A non-constant affine function (i.e., with vanishing Hessian) would fit this description. However, hyperbolic space does not admit such a function.

T. Sakai: "On Riemannian manifolds admitting a function whose gradient is of constant norm", Kodai Math. J. 19 (1996), 39-51.

Finally, a preprint of the paper is ready. https://www.researchgate.net/publication/291823768_ON_SOME_STRUCTURE_RESULTS_FOR_GODEL-TYPE_SPACETIMES_Preprint