There are three types of so-called fundamental forms. The most important are the first and second (since the third can be expressed in terms of these). The fundamental
forms are extremely important and useful in determining the metric properties of
a surface, such as line element, area
element, normal curvature, Gaussian
curvature, and mean curvature. Let 
be a regular surface with
points in the tangent
space 
of 
.
Then the first fundamental form is the inner product of tangent vectors,
 |
(1)
|
For 
,
the second fundamental form is the symmetric
bilinear form on the tangent space 
,
 |
(2)
|
where 
is the shape operator. The third
fundamental form is given by
 |
(3)
|
The first and second fundamental forms satisfy
 |  |  |
(4)
|
 |  |  |
(5)
|
where 
is a regular patch and 
and 
are the partial derivatives of 
with respect to parameters 
and 
, respectively. Their ratio is simply the normal
curvature
 |
(6)
|
for any nonzero tangent vector. The third fundamental form is given in terms of the first and second forms by
 |
(7)
|
where 
is the mean curvature and 
is the Gaussian curvature.
The first fundamental form (or line element) is given explicitly by the Riemannian metric
 |
(8)
|
It determines the arc length of a curve on a surface. The coefficients are given by
 |  |  |
(9)
|
 |  |  |
(10)
|
 |  |  |
(11)
|
The coefficients are also denoted 
, 
, and 
. In curvilinear
coordinates (where 
), the quantities
 |  |  |
(12)
|
 |  |  |
(13)
|
are called scale factors.
The second fundamental form is given explicitly by
 |
(14)
|
where
 |  |  |
(15)
|
 |  |  |
(16)
|
 |  |  |
(17)
|
and 
are the direction cosines of the surface normal.
The second fundamental form can also be written
 |  |  |
(18)
|
 |  |  |
(19)
|
 |  |  |
(20)
|
 |  |  |
(21)
|
 |  |  |
(22)
|
 |  |  |
(23)
|
 |  |  |
(24)
|
 |  |  |
(25)
|
where 
is the normal vector (Gray 1997, p. 368), or
 |  |  |
(26)
|
 |  |  |
(27)
|
 |  |  |
(28)
|
(Gray 1997, p. 379).
