Differential Geometry of Surfaces - Curvature of Surfaces in E3

Curvature of Surfaces in E3

See also: Gaussian curvature and Mean curvature

Informally Gauss defined the curvature of a surface in terms of the curvatures of certain plane curves connected with the surface. He later found a series of equivalent definitions. One of the first was in terms of the area-expanding properties of the Gauss map, a map from the surface to a 2-dimensional sphere. However, before obtaining a more intrinsic definition in terms of the area and angles of small triangles, Gauss needed to make an in-depth investigation of the properties of geodesics on the surface, i.e. paths of shortest length between two fixed points on the surface (see below).

The Gaussian curvature at a point on an embedded smooth surface given locally by the equation

z = F(x,y)

in E3, is defined to be the product of the principal curvatures at the point; the mean curvature is defined to be their average. The principal curvatures are the maximum and minimum curvatures of the plane curves obtained by intersecting the surface with planes normal to the tangent plane at the point. If the point is (0, 0, 0) with tangent plane z = 0, then, after a rotation about the z-axis setting the coefficient on xy to zero, F will have the Taylor series expansion

The principal curvatures are k₁ and k₂ in this case, the Gaussian curvature is given by

and the mean curvature by

Since K and K_m are invariant under isometries of E3, in general

and

where the derivatives at the point are given by P = F_x, Q = F_y, R = F_{x x}, S = F_{x y}, and T = F_{y y}.

For every oriented embedded surface the Gauss map is the map into the unit sphere sending each point to the (outward pointing) unit normal vector to the oriented tangent plane at the point. In coordinates the map sends (x,y,z) to

Direct computation shows that: the Gaussian curvature is the Jacobian of the Gauss map.