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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 ...
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]
where the second equality follows from the Gauss–Jacobi equation and the fourth from Gauss's derivative formula in the orthogonal coordinates (r,θ). Gauss's formula shows that the curvature at a point can be calculated as the limit of angle excess α + β + γ − π over area for successively smaller geodesic triangles near the point ...
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.
Gauss's Theorema egregium (Latin: "remarkable theorem") states that Gaussian curvature of a surface can be determined from the measurements of length on the surface itself. In fact, it can be found given the full knowledge of the first fundamental form and expressed via the first fundamental form and its partial derivatives of first and second ...
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 ...
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.
The equations of geodesics are easy to write in terms of the Riemannian connection, which can be locally expressed in terms of the Christoffel symbols. Along a curve in the surface, the connection defines a first order differential equation in the frame bundle.
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