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Einstein's new theory of gravitation is formulated in a geometric framework of curved spacetime. Here the subject of non-Euclidean geometry is introduced by way of Gauss's theory of curved surfaces. Generalized (Gaussian) coordinates: A systematic way to label points in space without reference to any objects outside this space. Metric function: For a given coordinate choice, the metric determines the intrinsic geometric properties of a curved space. Geodesic equation: It describes the shortest and the straightest possible curve in a warped space and is expressed in terms of the metric function. Curvature: It is a nonlinear second derivative of the metric. As the deviation from Euclidean relations is proportional to the curvature, it measures how much the space is warped.

This chapter deals with Riemann surfaces, coverings, and hypergeometric functions. It first considers the genus and Euler number of a Riemann surface before discussing Möbius transformations and notes that an automorphism of a Riemann surface is a biholomorphic map of the Riemann surface onto itself. It then describes a Riemannian metric and the Gauss-Bonnet theorem, which can be interpreted as a relation between the Gaussian curvature of a compact Riemann surface X and its Euler characteristic. It also examines the behavior of the Euler number under finite covering, along with finite subgroups of the group of fractional linear transformations PSL(2, C). Finally, it presents some basic facts about the classical Gauss hypergeometric functions of one complex variable, triangle groups acting discontinuously on one of the simply connected Riemann surfaces, and the hypergeometric monodromy group.

Two-dimensional (2D) electron systems on liquid helium and in semiconductor heterostructures are described. The quantum Hall effect is described as a distinguishing feature of a 2D electron system. Crystallinity is defined. The thermal vibrations of atoms in a 2D system are examined. The relative motions of atoms in 2D are regular and are used to define a melting criterion. Data are shown for Graphene in such a 2D melting criterion. Electrons contained in graphene are 2D, but graphene itself is 2D-3, permitting flexure into the third direction. Curvature and Gaussian curvature are defined. Poisson ratio, Lame constants and Young’s modulus are defined. Waves and vibrations on elastic plates and beams such as graphene are described. Crumpling of a membrane. The crumpling formula of de Gennes and Taupin, as applied to graphene. A comparison of one dimensional to two-dimensional systems is made. Thermal oscillations of graphene squares are estimated.