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

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

The coefficients are also denoted , , and . In curvilinear
coordinates (where ), the quantities

are called scale factors .

The second fundamental form is given explicitly by

(14)

where

and
are the direction cosines of the surface normal.
The second fundamental form can also be written

where
is the normal vector (Gray 1997, p. 368), or

(Gray 1997, p. 379).

See also Arc Length ,

Area Element ,

First Fundamental Form ,

Gaussian Curvature ,

Geodesic ,

Kähler Manifold ,

Line
of Curvature ,

Line Element ,

Mean
Curvature ,

Normal Curvature ,

Riemannian
Metric ,

Scale Factor ,

Second
Fundamental Form ,

Surface Area ,

Third
Fundamental Form ,

Weingarten Equations
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References Gray, A. "The Three Fundamental Forms." §16.6 in Modern
Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca
Raton, FL: CRC Press, pp. 368-371 and 380-382, 1997. Referenced on
Wolfram|Alpha Fundamental Forms
Cite this as:
Weisstein, Eric W. "Fundamental Forms."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/FundamentalForms.html

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