Page 343 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes constant mean curvature. Unlike Gauss curvature, the mean curvature is extrinsic and depends
on the embedding, for instance, a cylinder and a plane are locally isometric but the mean
curvature of a plane is zero while that of a cylinder is non-zero.

27.2.3 Second Fundamental Form

The intrinsic and extrinsic curvature of a surface can be combined in the second fundamental
form. This is a quadratic form in the tangent plane to the surface at a point whose value at a
particular tangent vector X to the surface is the normal component of the acceleration of a curve
along the surface tangent to X; that is, it is the normal curvature to a curve tangent to X.
Symbolically,


II(X,X) N ( X X)

where N is the unit normal to the surface. For unit tangent vectors X, the second fundamental
form assumes the maximum value k and minimum value k , which occur in the principal
2
1
directions u and u , respectively. Thus, by the principal axis theorem, the second fundamental
2
1
form is
2
2



II(X,X) k (X u )  k (X u ) .
1
2
2
1
Thus, the second fundamental form encodes both the intrinsic and extrinsic curvatures.
A related notion of curvature is the shape operator, which is a linear operator from the tangent
plane to itself. When applied to a tangent vector X to the surface, the shape operator is the
tangential component of the rate of change of the normal vector when moved along a curve on
the surface tangent to X. The principal curvatures are the eigenvalues of the shape operator, and
in fact the shape operator and second fundamental form have the same matrix representation
with respect to a pair of orthonormal vectors of the tangent plane. The Gauss curvature is, thus,
the determinant of the shape tensor and the mean curvature is half its trace.
27.3 Higher Dimensions: Curvature of Space
By extension of the former argument, a space of three or more dimensions can be intrinsically
curved; the full mathematical description is described at curvature of Riemannian manifolds.
Again, the curved space may or may not be conceived as being embedded in a higher-dimensional
space.
After the discovery of the intrinsic definition of curvature, which is closely connected with
non-Euclidean geometry, many mathematicians and scientists questioned whether ordinary
physical space might be curved, although the success of Euclidean geometry up to that time
meant that the radius of curvature must be astronomically large. In the theory of general relativity,
which describes gravity and cosmology, the idea is slightly generalised to the “curvature of
space-time”; in relativity theory space-time is a pseudo-Riemannian manifold. Once a time
coordinate is defined, the three-dimensional space corresponding to a particular time is generally
a curved Riemannian manifold; but since the time coordinate choice is largely arbitrary, it is the
underlying space-time curvature that is physically significant.

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

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