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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]
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 ...
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.
The exterior derivative is defined to be the unique ℝ-linear mapping from k-forms to (k + 1)-forms that has the following properties: The operator d {\displaystyle d} applied to the 0 {\displaystyle 0} -form f {\displaystyle f} is the differential d f {\displaystyle df} of f {\displaystyle f}
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 ...
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 .