Page 342 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 27: Principal Curvatures

Taking all possible tangent vectors then the maximum and minimum values of the normal Notes
curvature at a point are called the principal curvatures, k and k , and the directions of the
1
2
corresponding tangent vectors are called principal directions.
27.2 Two Dimensions: Curvature of Surfaces

27.2.1 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, k k . It has the dimension of 1/length and is positive for
2
1 2
spheres, negative for one-sheet hyperboloids and zero for planes. It determines whether a
surface is locally convex (when it is positive) or locally saddle (when it is negative).
This definition of Gaussian curvature is extrinsic in that it uses the surface’s embedding in R ,
3
normal vectors, external planes etc. Gaussian curvature is, however, in fact 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 which is a purely extrinsic type of curvature.
Formally, Gaussian curvature only depends on the Riemannian metric of the surface. This is
Gauss’s celebrated Theorema Egregium, which he found while concerned with geographic surveys
and map-making.

An intrinsic definition of the Gaussian curvature at a point P is the following: imagine an ant
which is tied to P with a short thread of length r. She runs around P while the thread is completely
stretched and measures the length C(r) of one complete trip around P. If the surface were flat, she
would find C(r) = 2. On curved surfaces, the formula for C(r) will be different, and the Gaussian
curvature K at the point P can be computed by the Bertrand–Diquet–Puiseux theorem as
3
K  lim(2 r C(r)) . .


r 0 r  3
The integral of the Gaussian curvature over the whole surface is closely related to the surface’s
Euler characteristic.
The discrete analog of curvature, corresponding to curvature being concentrated at a point and
particularly useful for polyhedra, is the (angular) defect; the analog for the Gauss-Bonnet theorem
is Descartes’ theorem on total angular defect.
Because 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.

27.2.2 Mean Curvature

The mean curvature is equal to half the sum of the principal curvatures, (k +k )/2. It has the
2
1
dimension of 1/length. Mean curvature is closely related to the first variation of surface area, in
particular a minimal surface such as a soap film, has mean curvature zero and a soap bubble has

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