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Thus the formula of the preceding article leads itself to the remarkable Theorem. If a curved surface is developed upon any other surface whatever, the measure of curvature in each point remains unchanged. Gauss's original statement of the Theorema Egregium, translated from Latin into English.
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 ...
The Gauss formula [6] now asserts that is the Levi-Civita connection for M, and is a symmetric vector-valued form with values in the normal bundle. It is often referred to as the second fundamental form. An immediate corollary is the Gauss equation for the curvature tensor.
Gauss's original statement of the Theorema Egregium, translated from Latin into English. Gauss's Theorema Egregium , the "Remarkable Theorem", shows that the Gaussian curvature of a surface can be computed solely in terms of the metric and is thus an intrinsic invariant of the surface, independent of any isometric embedding in E 3 and unchanged ...
This is called the Gauss equation, as it may be viewed as a generalization of Gauss's Theorema Egregium. For general Riemannian manifolds one has to add the curvature of ambient space; if N is a manifold embedded in a Riemannian manifold ( M , g ) then the curvature tensor R N of N with induced metric can be expressed using the second ...
In 1827, Carl Friedrich Gauss discovered that the Gaussian curvature of a surface embedded in 3-dimensional space only depends on local measurements made within the surface (the first fundamental form). [1] This result is known as the Theorema Egregium ("remarkable theorem" in Latin).
Theorema egregium; Gauss–Bonnet theorem; First fundamental form; Second fundamental form; Gauss–Codazzi–Mainardi equations; Dupin indicatrix; Asymptotic curve ...
This is Gauss's celebrated Theorema Egregium, which he found while concerned with geographic surveys and mapmaking. An intrinsic definition of the Gaussian curvature at a point P is the following: imagine an ant which is tied to P with a short thread of length r .