Editing Curvature (section)

===Gaussian curvature===
{{main|Gaussian curvature}}
In contrast to curves, which do not have intrinsic curvature, but do have extrinsic curvature (they only have a curvature given an embedding), surfaces can have intrinsic curvature, independent of an embedding. The [[Gaussian curvature]], named after [[Carl Friedrich Gauss]], is equal to the product of the principal curvatures, {{math|''k''<sub>1</sub>''k''<sub>2</sub>}}. It has a dimension of length<sup>−2</sup> and is positive for [[sphere]]s, negative for one-sheet [[hyperboloid]]s and zero for planes and [[Cylinder#Cylindrical surfaces|cylinders]]. It determines whether a surface is [[locally]] [[:wikt:convex|convex]] (when it is positive) or locally saddle-shaped (when it is negative).

Gaussian curvature is an ''intrinsic'' property of the surface, meaning it does not depend on the particular [[embedding]] of the surface; intuitively, this means that ants living on the surface could determine the Gaussian curvature. For example, an ant living on a sphere could measure the sum of the interior angles of a triangle and determine that it was greater than 180 degrees, implying that the space it inhabited had positive curvature. On the other hand, an ant living on a cylinder would not detect any such departure from [[Euclidean geometry]]; in particular the ant could not detect that the two surfaces have different mean curvatures (see below), which is a purely extrinsic type of curvature.

Formally, Gaussian curvature only depends on the [[Riemannian metric]] of the surface. This is [[Carl Friedrich Gauss|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 {{mvar|P}} is the following: imagine an ant which is tied to {{mvar|P}} with a short thread of length {{mvar|r}}. It runs around {{mvar|P}} while the thread is completely stretched and measures the length {{math|''C''(''r'')}} of one complete trip around {{mvar|P}}. If the surface were flat, the ant would find {{math|''C''(''r'') {{=}} 2π''r''}}. On curved surfaces, the formula for {{math|''C''(''r'')}} will be different, and the Gaussian curvature {{mvar|K}} at the point {{mvar|P}} can be computed by the [[Bertrand–Diguet–Puiseux theorem]] as

:<math> K = \lim_{r\to 0^+} 3\left(\frac{2\pi r-C(r)}{\pi r^3}\right).</math>

The [[integral]] of the Gaussian curvature over the whole surface is closely related to the surface's [[Euler characteristic]]; see the [[Gauss–Bonnet theorem]].

The discrete analog of curvature, corresponding to curvature being concentrated at a point and particularly useful for [[polyhedra]], is the [[defect (geometry)|(angular) defect]]; the analog for the [[Gauss–Bonnet theorem]] is [[Defect (geometry)#Descartes.27 theorem|Descartes' theorem on total angular defect]].

Because (Gaussian) curvature can be defined without reference to an embedding space, it is not necessary that a surface be embedded in a higher-dimensional space in order to be curved. Such an intrinsically curved two-dimensional surface is a simple example of a [[Riemannian manifold]].