The quantities in the denominator are intrinsic quantities. The quantities in the numerator depend on values in the 3-dimensional space in which the surface is embedded. What we now call the total curvature, or Gaussian curvature, may be defined as Gauss showed that could be expressed in terms of the coefficients in the first and second fundamental forms. Surface of (left) positive curvature and (right) negative curvature.

Gauss focussed on the product of the principal curvatures The principal curvatures, and, are the maximum and minimum values of, and they occur for planes in two orthogonal directions, the principal directions. If a vector is normal to the surface, any plane containing cuts the surface in a curve, with curvature. Gauss showed that the entire extrinsic geometry of the surface could be characterized by the second fundamental form. The functions, and are extrinsic: they depend upon the embedding space external to the surface, not just on values within the surface itself. The second fundamental form introduced by Gauss is The geometry may be Euclidean or non-Euclidean, or indeed a combination of these types, depending on the values of the coefficients in ( 1). It enables us to calculate the lengths of curves on the surface and the areas of regions on the surface. The first fundamental form ( 1) contains everything that is required to develop geometry on a two-dimensional surface or manifold. They are intrinsic quantities, independent of the space in which the surface is embedded. We note that the functions, and depend only upon values within the surface. By generalizing the theorem of Pythagoras, the line element on the surface can be writtenĮquation ( 1) is known as the first fundamental form of the surface. The cartesian coordinates are functions of these, so thatĪn obvious example is the latitude-longitude coordinate system on the surface of the spherical earth. Gauss introduced general coordinates to specify any point on a surface. Indeed, his new revelation was so dazzling that he could not but feel that others who claimed to do something important in the field of parallels trailed far behind him.’ The phrase which I have underlined here indicates the close link between the sign of the total curvature and the character of the geometry on the surface. This caused the normally restrained Gauss to use the adjective egregium, variously translated as outstanding, exceptional, extraordinary or amazing.Īs Lanczos (1970) put it, `In view of his customary reticence, it was an exceptionally jubilant gesture to call one of his theorems “Theorema egregium”. It is not necessary to have any information about the higher-dimensional space in which the surface is embedded. The Gaussian curvature can be calculated from measurements that the geometer can make within the surface. Imagine a geometer living on a two-dimensional surface, or manifold as Riemann called it. What is remarkable about Gauss’s theorem is that the total curvature is an intrinsic quantity. The Gaussian curvature characterizes the intrinsic geometry of a surface. The total curvature - or Gaussian curvature - depends only on measurements within the surface and Gauss showed that its value is independent of the coordinate system used.

He was inspired by his work on geodesy, surveying and map-making, which involved taking measurements on the surface of the Earth. Gauss defined a quantity that measures the curvature of a two-dimensional surface.

In 1828 he published his “Disquisitiones generales circa superficies curvas”, or General investigation of curved surfaces. One of greatest achievements of Carl Friedrich Gauss was a theorem so startling that he gave it the name Theorema Egregium or outstanding theorem. Surfaces of positive curvature (top), negative curvature (middle) and vanishing curvature (bottom).

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