1 Athanase Papadopoulos: Introduction -- 2 Norbert A'Campo and Athanase Papadopoulos, Geometry of surfaces -- 3 Ken’ichi Ohshika: Teichmuller spaces and their various metrics -- 4 Marc Troyanov: Double forms, curvature integrals and the Gauss-Bonnet formula -- 5 Graham Smith, Quaternions, Monge–Amp˜A¨re structures and κ-surfaces -- 6 Peter Kristel and Eric Schippers, Lagrangian Grassmannians of polarizations -- 7 Arpad Kurusa, Metric characterizations of projective-metric spaces -- 8 Arpad Kurusa, Supplement to “Metric Characterization of Projective-metric Spaces” -- 9 Boumediene Et-Taoui, Metric problems in projective and Grassmann spaces -- 10 Valeriı Berestovskiı and Yuriı Nikonorov, On the geometry of finite homogeneous subsets of Euclidean spaces -- 11 Gue-Seon Lee and Ludovic Marquis, Discrete Coxeter groups -- 12 Bruno Luiz Santos Correia and Marc Troyanov, Isoperimetry inFinitely Generated Groups.

The book is the second volume of a collection which consists of surveys that focus on important topics in geometry which are at the heart of current research. The topics in the present volume include the conformal and the metric geometry of surfaces, Teichmüller spaces, immersed surfaces of prescribed extrinsic curvature in 3-dimensional manifolds, symplectic geometry, the metric theory of Grassmann spaces, homogeneous metric spaces, polytopes, the higher-dimensional Gauss–Bonnet formula, isoperimetry in finitely generated groups and Coxeter groups. Each chapter is intended for graduate students and researchers. Several chapters are based on lectures given by their authors to middle-advanced level students and young researchers. The whole book is intended to be an introduction to important topics in geometry.