Page 345 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes Another generalization of curvature relies on the ability to compare a curved space with another
space that has constant curvature. Often this is done with triangles in the spaces. The notion of a
triangle makes senses in metric spaces, and this gives rise to CAT (k) spaces.

27.5 Summary

All curves with the same tangent vector will have the same normal curvature, which is the

same as the curvature of the curve obtained by intersecting the surface with the plane
containing T and u. Taking all possible tangent vectors then the maximum and minimum
values of the normal curvature at a point are called the principal curvatures, k and k , and
1
2
the directions of the corresponding tangent vectors are called principal directions.
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. 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. 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

The mean curvature is equal to half the sum of the principal curvatures, (k +k )/2. It has
 1 2
the 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 constant mean curvature. Although an arbitrarily-curved space is
very complex to describe, the curvature of a space which is locally isotropic and
homogeneous is described by a single Gaussian curvature, as for a surface; mathematically
these are strong conditions, but they correspond to reasonable physical assumptions (all
points and all directions are indistinguishable). A positive curvature corresponds to the
inverse square radius of curvature; an example is a sphere or hypersphere. An example of
negatively curved space is hyperbolic geometry. A space or space-time with zero curvature
is called flat. For example, Euclidean space is an example of a flat space, and Minkowski
space is an example of a flat space-time

27.6 Keywords

Principal directions: Taking all possible tangent vectors then the maximum and minimum
values of the normal curvature at a point are called the principal curvatures, k and k , and the
2
1
directions of the corresponding tangent vectors are called principal directions.
Gaussian curvature: 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.
Arbitrarily-curved space: An arbitrarily-curved space is very complex to describe, the curvature
of a space which is locally isotropic and homogeneous is described by a single Gaussian curvature,
as for a surface; mathematically these are strong conditions, but they correspond to reasonable
physical assumptions.

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