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''metric form, of a surface''

A quadratic form in the differentials of the coordinates on the surface that determines the intrinsic geometry of the surface in a neighbourhood of a given point.

Let the surface be defined by the equation

$$ 
\mathbf r  =  \mathbf r ( u , v),
$$

where  $  u $
and  $  v $
are coordinates on the surface, while

$$ 
d \mathbf r  =  \mathbf r _ {u}  du + \mathbf r _ {v}  dv
$$

is the differential of the position vector  $  \mathbf r $
along the chosen direction  $  du :dv $
of displacement from a point  $  M $
to an infinitesimally close point  $  M  ^  \prime  $(
see Fig. a).

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/f040450a.gif" />

Figure: f040450a

The square of the principal linear part of the increment of the length of the arc  $  MM  ^  \prime  $
can be expressed in terms of the square of the differential  $  d \mathbf r $:

$$ 
\textrm{ I }  =  ds  ^ {2}  =  d \mathbf r  ^ {2}  =  \mathbf r _ {u}  ^ {2}  du  ^ {2} + 2 \mathbf r _ {u} \mathbf r _ {v}  du  dv + \mathbf r _ {v}  ^ {2}  dv  ^ {2} ,
$$

and is called the first fundamental form of the surface. The coefficients in the first fundamental form are usually denoted by

$$ 
E  =  \mathbf r _ {u}  ^ {2} ,\ \ 
F  =  ( \mathbf r _ {u} , \mathbf r _ {v} ) ,\ \ 
G  =  \mathbf r _ {v}  ^ {2} ,
$$

or, in tensor symbols,

$$ 
d \mathbf r  ^ {2}  =  g _ {11}  du  ^ {2} + 2g _ {12}  du  dv + g _ {22}  dv  ^ {2} .
$$

The tensor  $  g _ {ij} $
is called the first fundamental, or metric, tensor of the surface. The first fundamental form is a positive-definite form at regular points on the surface:

$$ 
EG - F ^ { 2 }  >  0.
$$

The first fundamental form characterizes the metric properties of the surface: Knowledge of the first fundamental form enables one to calculate arc lengths on the surface:

$$ 
s  =  \int\limits _ {t _ {0} } ^ { t }  \sqrt {E \left ( 
\frac{du}{dt}
 \right )  ^ {2} + 2F 
\frac{du}{dt}
 
\frac{dv}{dt}
 + G \left ( 
\frac{dv}{dt}
 \right )  ^ {2} }  dt,
$$

where  $  t $
is a parameter on the curve; angles between curves on the surface:

$$ 
\cos  ( \widehat{ {dr  \delta r }}  ) =
$$

$$ 
= \ 

\frac{E  du  \delta u + F( du  \delta v+ dv  \delta u)+ G  dv  \delta v }{\sqrt {E  du
 ^ {2} + 2F  du  dv+ G  dv  ^ {2} } \sqrt {E  \delta u  ^ {2}
+ 2F  \delta u  \delta v+ G  \delta v  ^ {2} } }
 ,
$$

where  $  du : dv $
and  $  \delta u : \delta v $
are the directions of the tangent vectors to the curves (see Fig. b); and areas of regions on the surface:

$$ 
\sigma  =  \int\limits \int\limits \sqrt {EG - F ^ { 2 } }  du  dv .
$$

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/f040450b.gif" />

Figure: f040450b

The form of the coefficients of the first fundamental form substantially depends on the choice of the coordinates. The first fundamental form has so-called orthogonal form

$$ 
E( u, v)  du  ^ {2} + G( u, v)  dv  ^ {2}
$$

in orthogonal coordinates, canonical form

$$ 
du  ^ {2} + G  ^ {2}  dv  ^ {2}
$$

in semi-geodesic coordinates, and isothermal (isometric) form in isothermal coordinates:

$$ 
ds  ^ {2}  =  \lambda  ^ {2} ( u, v)( du  ^ {2} + dv  ^ {2} ).
$$

Sometimes a surface is characterized by special forms of the first fundamental form. For example, a [[Liouville surface|Liouville surface]] is characterized by:

$$ 
[ \phi ( u) + \psi ( v)]( du  ^ {2} + dv  ^ {2} ).
$$

The first fundamental form is a bending invariant for the surface: The Gaussian curvature at a given point can be calculated from the coefficients of the first fundamental form and their derivatives only (Gauss' theorem).

See [[Fundamental forms of a surface|Fundamental forms of a surface]] for the relation between the first fundamental form and other quadratic forms, as well as for literature.

====Comments====
In more current terminology the first fundamental form of an imbedded surface  $  f:  U \rightarrow \mathbf R  ^ {3} $, 
$  U \subset  \mathbf R  ^ {2} $, 
is defined as follows. The inclusion  $  ( Tf  ) _ {u} ( \mathbf R _ {u}  ^ {2} ) \subset  T _ {f ( u) }  \mathbf R  ^ {3} $
defines a quadratic form on the tangent space  $  ( Tf  ) _ {u} ( \mathbf R _ {u}  ^ {2} ) $
to the surface at  $  f ( u) $
by restricting the canonical scalar product on  $  T _ {f ( u) }  \mathbf R  ^ {3} \simeq \mathbf R  ^ {3} $. 
Thus the first fundamental form of the surface  $  f ( U) \subset  \mathbf R  ^ {3} $
simply assigns to a pair of tangent vectors  $  X, Y $
to  $  f ( U) $
at  $  z \in f ( U) $
the [[Inner product|inner product]] of  $  X $
and  $  Y $
in  $  \mathbf R  ^ {3} $.

Gauss' theorem is also known as the theorem egregium, cf. [[Gauss theorem|Gauss theorem]].

The first fundamental form is also defined in higher (co)dimensions and for Riemannian spaces as ambient spaces (cf. [[#References|[a1]]]–[[#References|[a3]]]). In the last case it is given by the restriction of the [[Riemannian metric|Riemannian metric]] to the tangent subspaces defined by the given submanifold.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , Springer (1973) {{MR|0350630}} {{ZBL|0264.53001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> N.J. Hicks, "Notes on differential geometry" , v. Nostrand (1965) {{MR|0179691}} {{ZBL|0132.15104}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , '''1–2''' , Interscience (1963–1969) {{MR|1393941}} {{MR|1393940}} {{MR|0238225}} {{MR|1533559}} {{MR|0152974}} {{ZBL|0526.53001}} {{ZBL|0508.53002}} {{ZBL|0175.48504}} {{ZBL|0119.37502}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> C.C. Hsiung, "A first course in differential geometry" , Wiley (1981) pp. Chapt. 3, Sect. 4 {{MR|0608028}} {{ZBL|0458.53001}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> R.S. Millman, G.D. Parker, "Elements of differential geometry" , Prentice-Hall (1977) pp. 26 {{MR|0442832}} {{ZBL|0425.53001}} </TD></TR></table>

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