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Isometric embeddings of the reduced horizon of a Kerr-Newman black hole into Euclidean 3-space

We prove some results concerning Killing fields, isometric immersions and black hole horizons, and then combine them to arrive at conclusions concerning embeddings of reduced black hole horizons into Euclidean space. We show that not far from the Schwarzschild solution in the class of Kerr-Newman solutions of Einstein equations the reduced horizon of a black hole has a unique isometric embedding in . But certain critical values of parameters describing this class exist, for which it does not take place; when the reduced horizon has negative Gauss curvature areas. We compare these parameters with the thermodynamics of black holes.