This group gathers theorems from differential geometry, the study of curved surfaces and manifolds using the tools of calculus. It covers the intrinsic curvature results of Gauss and Gauss-Bonnet that measure how a surface bends independently of how it sits in space, the theory of geodesics and comparison theorems, including the Hopf-Rinow, Bonnet-Myers, Cartan-Hadamard, and Rauch comparison theorems, that describe shortest paths on curved spaces, and the Nash embedding theorem showing that every such abstract curved space can be realized concretely inside ordinary Euclidean space.
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Era Of Emergence All Differential Geometry
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1. Differential Geometry (Wikipedia)
WikipediaIntrinsic geometry and non-Euclidean geometry (1800-1900)Quote, Intrinsic geometry and non-Euclidean geometry (1800-1900)
The field of differential geometry became an area of study considered in its own right, distinct from the more broad idea of analytic geometry, in the 1800s, primarily through the foundational work of Carl Friedrich Gauss and Bernhard Riemann, and also in the important contributions of Nikolai Lobachevsky on hyperbolic geometry and non-Euclidean geometry and throughout the same period the development of projective geometry.
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