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In differential geometry, the Gauss map of a surface is a function that maps each point in the surface to a unit vector that is orthogonal to the surface at that point. Namely, given a surface X in Euclidean space R 3 , the Gauss map is a map N : X → S 2 (where S 2 is the unit sphere ) such that for each p in X , the function value N ( p ) is ...
The differential dn of the Gauss map n can be used to define a type of extrinsic curvature, known as the shape operator [55] or Weingarten map. This operator first appeared implicitly in the work of Wilhelm Blaschke and later explicitly in a treatise by Burali-Forti and Burgati. [56]
Moreover, the Gauss map of M into S 2 induces a natural map between the associated frame bundles which is equivariant for the actions of SO(2). [ 29 ] Cartan's idea of introducing the frame bundle as a central object was the natural culmination of the theory of moving frames , developed in France by Darboux and Goursat .
Gauss's original statement of the Theorema Egregium, translated from Latin into English. The theorem is "remarkable" because the definition of Gaussian curvature makes ample reference to the specific way the surface is embedded in 3-dimensional space, and it is quite surprising that the result does not depend on its embedding.
In 1822, Carl Friedrich Gauss proved the existence of isothermal coordinates on an arbitrary surface with a real-analytic Riemannian metric, following earlier results of Joseph Lagrange in the special case of surfaces of revolution. [1]
differentiable map. submersion; immersion; Embedding. Whitney embedding theorem; Critical value. Sard's theorem; Saddle point; Morse theory; Lie derivative; Hairy ball theorem; Poincaré–Hopf theorem; Stokes' theorem; De Rham cohomology; Sphere eversion; Frobenius theorem (differential topology) Distribution (differential geometry) integral ...
In Riemannian geometry, Gauss's lemma asserts that any sufficiently small sphere centered at a point in a Riemannian manifold is perpendicular to every geodesic through the point. More formally, let M be a Riemannian manifold, equipped with its Levi-Civita connection, and p a point of M. The exponential map is a mapping from the tangent space ...
In Riemannian geometry and pseudo-Riemannian geometry, the Gauss–Codazzi equations (also called the Gauss–Codazzi–Weingarten-Mainardi equations or Gauss–Peterson–Codazzi formulas [1]) are fundamental formulas that link together the induced metric and second fundamental form of a submanifold of (or immersion into) a Riemannian or pseudo-Riemannian manifold.