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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 ...
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 under coordinate transformations. In particular, isometries and local isometries of ...
Gauss's Theorema Egregium (differential geometry) Gauss–Bonnet theorem ... Theorem of de Moivre–Laplace (probability theory) Theorem of the cube (algebraic varieties)
As a result, the Theorema Egregium (remarkable theorem), established a property of the notion of Gaussian curvature. Informally, the theorem says that the curvature of a surface can be determined entirely by measuring angles and distances on the surface, regardless of the embedding of the surface in three-dimensional or two-dimensional space. [197]
Theorem: Every smooth manifold admits a (non-canonical) Riemannian metric. [13] This is a fundamental result. Although much of the basic theory of Riemannian metrics can be developed using only that a smooth manifold is a locally Euclidean topological space, for this result it is necessary to use that smooth manifolds are Hausdorff and paracompact.
This is also a consequence of Carl Gauss's 1827 Theorema Egregium [Remarkable Theorem]. A conformal parameterization of a disc-like domain on the sphere is deemed scale-optimal when it minimizes the ratio of maximum to minimum scale across the entire map. This occurs by assigning a unit scale to the boundary of the disc.
Theorema is also Latin for "theorem", and a number of theorems are sometimes referred to by a Latin name in English, most notably two theorems of Carl Friedrich Gauss: Theorema Egregium , "Remarkable Theorem", best-known example