Special geometry, black holes and Euclidean supersymmetry,” arXiv:hep-th/0703037

We review recent developments in special geometry and explain its role in the theory of supersymmetric black holes. To make this article self-contained, a short introduction to black holes is given, with emphasis on the laws of black hole mechanics and black hole entropy. We also summarize the existing results on the para-complex version of special geometry, which occurs in Euclidean supersymmetry. The role of real coordinates in special geometry is illustrated, and we briefly indicate how Euclidean supersymmetry can be used to study stationary black hole solutions via dimensional reduction over time. This article is an updated and substantially extended version of the previous review article ‘New developments in special geometry’, hep-th/0602171. Contribution to the Handbook on Pseudo-Riemannian Geometry and Supersymmetry, ed. V. Cortés, published by the European Mathematical Society in the series “IRMA Lectures in Mathematical and Theoretical Physics.”