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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 ...
In the language of tensor calculus, making use of natural metrics and connections on tensor bundles, the Gauss equation can be written as H 2 − |h| 2 = R and the two Codazzi equations can be written as ∇ 1 h 12 = ∇ 2 h 11 and ∇ 1 h 22 = ∇ 2 h 12; the complicated expressions to do with Christoffel symbols and the first fundamental form ...
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 ...
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. An explicit expression for the Gaussian curvature in terms of the first fundamental form is provided by the Brioschi formula.
The Weingarten equation is an analog of the Gauss formula for a connection in the normal bundle. Let X ∈ T M {\displaystyle X\in TM} and ξ {\displaystyle \xi } a normal vector field. Then decompose the ambient covariant derivative of ξ {\displaystyle \xi } along X into tangential and normal components:
It is named after the mathematician Carl Friedrich Gauss. The graph of a Gaussian is a characteristic symmetric " bell curve " shape. The parameter a is the height of the curve's peak, b is the position of the center of the peak, and c (the standard deviation , sometimes called the Gaussian RMS width) controls the width of the "bell".
This is a list of formulas encountered in Riemannian geometry. Einstein notation is used throughout this article. This article uses the "analyst's" sign convention for Laplacians, except when noted otherwise.