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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]
In this context, the first equation, often called the Gauss equation (after its discoverer Carl Friedrich Gauss), says that the Gauss curvature of the surface, at any given point, is dictated by the derivatives of the Gauss map at that point, as encoded by the second fundamental form. [2]
In the mathematical field of differential geometry, the Osserman–Xavier–Fujimoto theorem concerns the Gauss maps of minimal surfaces in the three-dimensional Euclidean space. It says that if a minimal surface is immersed and geodesically complete , then the image of the Gauss map either consists of a single point (so that the surface is a ...
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.
where is the Gauss map, and the differential of regarded as a vector-valued differential form, and the brackets denote the metric tensor of Euclidean space. More generally, on a Riemannian manifold, the second fundamental form is an equivalent way to describe the shape operator (denoted by S ) of a hypersurface,
Tyler. Another name that exploded in popularity during the 1990s, Tyler is an English name with a literal meaning: "maker of tiles." In the 1990s, just over 262,000 Tylers were born in the United ...
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 .