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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 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 ...
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 ...
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; ... Einstein–Cartan theory; connection (vector bundle) connection ...
This is an accepted version of this page This is the latest accepted revision, reviewed on 8 January 2025. German mathematician, astronomer, geodesist, and physicist (1777–1855) "Gauss" redirects here. For other uses, see Gauss (disambiguation). Carl Friedrich Gauss Portrait by Christian Albrecht Jensen, 1840 (copy from Gottlieb Biermann, 1887) Born Johann Carl Friedrich Gauss (1777-04-30 ...
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) Gelfond–Schneider theorem (transcendental number theory)