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The Gauss map provides a mapping from every point on a curve or a surface to a corresponding point on a unit sphere. In this example, the curvature of a 2D-surface is mapped onto a 1D unit circle. 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 ...
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]
The sum of indices of the zeroes of the old (and new) vector field is equal to the degree of the Gauss map from the boundary of N ε to the (n–1)-dimensional sphere. Thus, the sum of the indices is independent of the actual vector field, and depends only on the manifold M.
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 ...
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 ...
Below are some examples of how differential geometry is applied to other fields of science and mathematics. In physics, differential geometry has many applications, including: Differential geometry is the language in which Albert Einstein's general theory of relativity is expressed.
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 differential topology, sphere eversion is the process of turning a sphere inside out in a three-dimensional space (the word eversion means "turning inside out"). It is possible to smoothly and continuously turn a sphere inside out in this way (allowing self-intersections of the sphere's surface) without cutting or tearing it or creating any ...