Quaternion-Kähler spaces, hyper-Kähler cones, and the c-map / Martin Roček, Cumrun Vafa, Stefan Vandoren -- Differential forms on quaternionic Kähler manifolds / Gregor Weingart -- Sasakian geometry, holonomy, and supersymmetry / Charles P. Boyer, Krzysztof Galicki† -- Special geometry for arbitrary signatures / María A. Lledó, Óscar Maciá, Antoine Van Proeyen, Veeravalli S. Varadarajan -- Special geometry, black holes and Euclidean supersymmetry / Thomas Mohaupt -- Generalized geometry – an introduction / Nigel Hitchin -- Generalizing geometry – algebroids and sigma models / Alexei Kotov, Thomas Strobl -- A potential for generalized Kähler geometry / Ulf Lindström, Martin Roček, Rikard von Unge, Maxim Zabzine -- Non-integrable geometries, torsion, and holonomy / Ilka Agricola -- Connections with totally skew-symmetric torsion and nearly-Kähler geometry / Paul-Andi Nagy -- Homogeneous nearly Kähler manifolds / Jean-Baptiste Butruille -- Nearly pseudo-Kähler and nearly para-Kähler six-manifolds / Lars Schäfer, Fabian Schulte-Hengesbach -- Quaternionic geometries from superconformal symmetry / Andrew Swann -- Twistor and reflector spaces of almost para-quaternionic manifolds / Stefan Ivanov, Ivan Minchev, Simeon Zamkovoy -- Para-pluriharmonic maps and twistor spaces / Matthias Krahe -- Maximally homogeneous para-CR manifolds of semisimple type / Dmitri V. Alekseevsky, Costantino Medori, Adriano Tomassini -- Recent developments in pseudo-Riemannian holonomy theory / Anton Galaev, Thomas Leistner -- Geometric applications of irreducible representations of Lie groups / Antonio J. Di Scala, Thomas Leistner, Thomas Neukirchner -- Surface holonomy / Konrad Waldorf -- Classification results for pseudo-Riemannian symmetric spaces / Ines Kath -- Pseudo-Kähler and para-Kähler symmetric spaces / Dmitri V. Alekseevsky -- Prehomogeneous affine representations and flat pseudo-Riemannian manifolds / Oliver Baues -- The conformal analog of Calabi–Yau manifolds / Helga Baum -- Nondegenerate conformal structures, CR structures and quaternionic CR structures on manifolds / Yoshinobu Kamishima -- Linear wave equations on Lorentzian manifolds / Christian Bär -- Survey of D-branes and K-theory / Daniel S. Freed.

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The purpose of this handbook is to give an overview of some recent developments in differential geometry related to supersymmetric field theories. The main themes covered are: special geometry and supersymmetry generalized geometry geometries with torsion para-geometries holonomy theory symmetric spaces and spaces of constant curvature conformal geometry wave equations on Lorentzian manifolds D-branes and K-theory The intended audience consists of advanced students and researchers working in differential geometry, string theory and related areas. The emphasis is on geometrical structures occurring on target spaces of supersymmetric field theories. Some of these structures can be fully described in the classical framework of pseudo-Riemannian geometry. Others lead to new concepts relating various fields of research, such as special Kähler geometry or generalized geometry.