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The [[Normal curvature|normal curvature]] of a surface in a principal direction, i.e. in a direction in which it assumes an extremal value. The principal curvatures  $  k _ {1} $
and  $  k _ {2} $
are the roots of the quadratic equation

$$ \tag{* }
\left |

\begin{array}{ll}
L - kE  &M - kF  \\
M - kF  &N - kG  \\
\end{array}

\right |  =  0,
$$

where  $  E $, 
$  F $
and  $  G $
are the coefficients of the [[First fundamental form|first fundamental form]], while  $  L $, 
$  M $
and  $  N $
are the coefficients of the [[Second fundamental form|second fundamental form]] of the surface, computed at the given point.

The half-sum of the principal curvatures  $  k _ {1} $
and  $  k _ {2} $
of the surface gives the [[Mean curvature|mean curvature]], while their product is equal to the [[Gaussian curvature|Gaussian curvature]] of the surface. Equation (*) may be written as

$$ 
k  ^ {2} - 2Hk + K  =  0,
$$

where  $  H $
is the mean, and  $  K $
is the Gaussian curvature of the surface at the given point.

The principal curvatures  $  k _ {1} $
and  $  k _ {2} $
are connected with the normal curvature  $  \widetilde{k}  $, 
taken in an arbitrary direction, by means of Euler's formula:

$$ 
\widetilde{k}   =  k _ {1}  \cos  ^ {2}  \phi + k _ {2}  \sin  ^ {2}  \phi ,
$$

where  $  \phi $
is the angle formed by the selected direction with the principal direction for  $  k _ {1} $.

====Comments====
In the case of an  $  m $-
dimensional submanifold  $  M $
of Euclidean  $  n $-
space  $  E  ^ {n} $
principal curvatures and principal directions are defined as follows.

Let  $  \xi $
be a unit normal to  $  M $
at  $  p \in M $. 
The Weingarten mapping (shape operator)  $  A _  \xi  $
of  $  M $
at  $  p $
in direction  $  \xi $
is given by the tangential part of  $  - \overline \nabla \; _ {\overline \xi \; }  $, 
where  $  \overline \nabla \; $
is the [[Covariant differential|covariant differential]] in  $  E  ^ {n} $
and  $  \overline \xi \; $
is a local extension of  $  \xi $
to a unit normal vector field.  $  A _  \xi  $
does not depend on the chosen extension of  $  \xi $. 
The principal curvatures of  $  M $
at  $  p $
in direction  $  \xi $
are given by the eigen values of  $  A _  \xi  $, 
the principal directions by its eigen directions. The (normalized) elementary symmetric functions of the eigen values of  $  A _  \xi  $
define the higher mean curvatures of  $  M $, 
which include as extremal cases the mean curvature as the trace of  $  A _  \xi  $
and the Lipschitz–Killing curvature as its determinant.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.J. Hicks,   "Notes on differential geometry" , v. Nostrand  (1965)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B.-Y. Chen,   "Geometry of submanifolds" , M. Dekker  (1973)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Berger,   B. Gostiaux,   "Differential geometry: manifolds, curves, and surfaces" , Springer  (1988)  (Translated from French)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H.S.M. Coxeter,   "Introduction to geometry" , Wiley  (1963)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M.P. Do Carmo,   "Differential geometry of curves and surfaces" , Prentice-Hall  (1976)  pp. 145</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  H.W. Guggenheimer,   "Differential geometry" , McGraw-Hill  (1963)  pp. 25; 60</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  D. Hilbert,   S.E. Cohn-Vossen,   "Geometry and the imagination" , Chelsea  (1952)  (Translated from German)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  B. O'Neill,   "Elementary differential geometry" , Acad. Press  (1966)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  M. Spivak,   "A comprehensive introduction to differential geometry" , '''1979''' , Publish or Perish  pp. 1–5</TD></TR></table>