In mathematics, the differential geometry of surfaces deals with smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: extrinsically, relating to their embedding in Euclidean space and intrinsically, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss (1825-1827), who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space.
Surfaces naturally arise as graphs of functions of a pair of variables, and sometimes appear in parametric form or as loci associated to space curves. An important role in their study has been played by Lie groups (in the spirit of the Erlangen program), namely the symmetry groups of the Euclidean plane, the sphere and the hyperbolic plane. These Lie groups can be used to describe surfaces of constant Gaussian curvature; they also provide an essential ingredient in the modern approach to intrinsic differential geometry through connections. On the other hand extrinsic properties relying on an embedding of a surface in Euclidean space have also been extensively studied. This is well illustrated by the non-linear Euler-Lagrange equations in the calculus of variations: although Euler developed the one variable equations to understand geodesics, defined independently of an embedding, one of Lagrange's main applications of the two variable equations was to minimal surfaces, a concept that can only be defined in terms of an embedding.
Read more about Differential Geometry Of Surfaces: Overview, History of Surfaces, Curvature of Surfaces in E3, Local Metric Structure, Geodesic Curves On A Surface, Geodesic Polar Coordinates, Gauss–Bonnet Theorem, Surfaces of Constant Curvature, Surfaces of Non-positive Curvature, Riemannian Connection and Parallel Transport, Global Differential Geometry of Surfaces, Reading Guide
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... sophisticated graduate-level treatments using the Riemannian connection on a surface can be found in Singer Thorpe (1967), do Carmo (1976) and O'Neill ...
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