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We investigate the general geometric properties of the surface of infinitered-shift corresponding to the event horizon of the static and axisymmetricsolution of the Einstein vacuum equations that only possesses mass $M$ andquadrupole moment $Q$. The deformation of the Schwarzschild surface $r=2M$produced by the quadrupole moment is shown, and the range of values of thismultipole moment is specified, which preserves a regular, closed, continuousand differentiable surface. Some thermodynamic consequences and speculationsensuing from our results are discussed.

Event Horizon of the Monopole-Quadrupole solution: geometric and thermodynamic properties