Surface Geometry of Charged Rotating Black Holes

Invariant measures of the surface geometry of a charged rotating (Kerr-Newman) black hole are examined. It is shown that as the rotation rate of the black hole increases, the equatorial circumference increases while the polar circumference decreases. This is analogous to effects in material rotating bodies. The number of parameters describing a charged Kerr black hole drops from three to two on its surface. It is found that a scale parameter $\eta$ and a distortion parameter $\beta$ describe this geometry very simply. There emerge two classes of Kerr metrics separated by $\beta =\frac{1}{2}$. For larger $\beta$ the Gaussian curvature becomes negative on two polar-cap regions and the surface cannot be globally embedded in Euclidean 3-space. Possible physical effects are briefly discussed.