The product of the principal curvatures (cf.
Principal curvature)
of a regular surface at a given point.
If
is the
first fundamental form
of the surface and
is the
second fundamental form
of the surface, then the Gaussian curvature can be computed by the formula
The Gaussian curvature is identical with the Jacobi determinant of the
spherical map:
where
is a point on the surface,
is the area of a domain
which contains
,
is the area of the spherical image of
,
and
is the diameter of the domain. The Gaussian curvature is positive at an
elliptic point,
negative at a
hyperbolic point,
and is zero at a
parabolic point
or a
flat point.
It may be expressed in terms of the coefficients of
the first fundamental form and their derivatives alone (the
Gauss theorem),
viz.
where
Since the Gaussian curvature depends on the metric only, i.e.
on the coefficients of the first fundamental form,
the Gaussian curvature is invariant under isometric deformation (cf.
Deformation, isometric).
The Gaussian curvature plays a special role in the theory
of surfaces, and many formulas are available for its computation,
[2].
The concept was introduced by
C.F. Gauss
[1],
and was named after him.