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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 ...
The crowning result, the Theorema Egregium of Gauss, established that the Gaussian curvature is an intrinsic invariant, i.e. invariant under local isometries. This point of view was extended to higher-dimensional spaces by Riemann and led to what is known today as Riemannian geometry. The nineteenth century was the golden age for the theory of ...
However, the Theorema Egregium of Carl Friedrich Gauss showed that for surfaces, the existence of a local isometry imposes that the Gaussian curvatures at the corresponding points must be the same. In higher dimensions, the Riemann curvature tensor is an important pointwise invariant associated with a Riemannian manifold that measures how close ...
Theorema egregium; Gauss–Bonnet theorem; First fundamental form; Second fundamental form; Gauss–Codazzi–Mainardi equations; Dupin indicatrix; Asymptotic curve ...
Gauss theorem (vector calculus) Gamas's Theorem (multilinear algebra) Gauss's Theorema Egregium (differential geometry) Gauss–Bonnet theorem (differential geometry) Gauss–Lucas theorem (complex analysis) Gauss–Markov theorem ; Gauss–Wantzel theorem ; Gelfand–Mazur theorem (Banach algebra) Gelfand–Naimark theorem (functional analysis)
Theorema egregium of Gauss states that the Gaussian curvature of a surface can be expressed solely in terms of the first fundamental form and its derivatives, so that K is in fact an intrinsic invariant of the surface.
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.