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A method in [[Differential geometry|differential geometry]] for the local examination of submanifolds of various homogeneous spaces, in which the starting point is to provide the submanifold itself and all its geometrical objects with the most general possible (mobile) frame (of reference). This method includes making the frame of reference canonical, namely assigning to each point in the submanifold a unique frame of reference in an invariant manner, in order to obtain differential invariants characterizing the submanifold apart from transformations imbedding it in the surrounding [[Homogeneous space|homogeneous space]]. This method was proposed in its most general form by E. Cartan [[#References|[1]]], who gave various examples of its application. Subsequently, the method was widely used and developed (see [[Method of extensions and restrictions|Method of extensions and restrictions]]). The analytic basis of the method is constituted by the invariant linear differential forms of Lie groups and their structure equations, as well as by the theory of representations of Lie groups as transformation groups. In modern geometry, the basic concepts of the method have required refinement, and they have been formulated in terms of the theory of bundles.
 
A method in [[Differential geometry|differential geometry]] for the local examination of submanifolds of various homogeneous spaces, in which the starting point is to provide the submanifold itself and all its geometrical objects with the most general possible (mobile) frame (of reference). This method includes making the frame of reference canonical, namely assigning to each point in the submanifold a unique frame of reference in an invariant manner, in order to obtain differential invariants characterizing the submanifold apart from transformations imbedding it in the surrounding [[Homogeneous space|homogeneous space]]. This method was proposed in its most general form by E. Cartan [[#References|[1]]], who gave various examples of its application. Subsequently, the method was widely used and developed (see [[Method of extensions and restrictions|Method of extensions and restrictions]]). The analytic basis of the method is constituted by the invariant linear differential forms of Lie groups and their structure equations, as well as by the theory of representations of Lie groups as transformation groups. In modern geometry, the basic concepts of the method have required refinement, and they have been formulated in terms of the theory of bundles.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m0650901.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m0650902.png" />-dimensional homogeneous space and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m0650903.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m0650904.png" />-dimensional [[Lie group|Lie group]] of its transformations (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m0650905.png" /> acts from the left). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m0650906.png" /> be a representation, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m0650907.png" /> is the [[Isotropy group|isotropy group]] (stationary group) of a certain point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m0650908.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m0650909.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509011.png" />, be a basis of left-invariant vector fields on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509012.png" /> such that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509013.png" /> restricted to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509014.png" /> also constitute a basis of left-invariant vector fields for the Lie subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509015.png" />. The basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509016.png" /> corresponds to a dual basis of left-invariant linear differential forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509017.png" /> on the Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509018.png" />. The canonical projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509019.png" /> puts the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509020.png" /> into correspondence with the left cosets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509021.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509022.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509023.png" />, and it introduces the structure of a principal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509024.png" />-bundle with base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509025.png" /> and structure group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509026.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509027.png" /> on the Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509028.png" />. With this representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509029.png" />, the vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509030.png" /> constitute a basis of fundamental vector fields for the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509031.png" />, while the vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509032.png" /> span a certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509033.png" />-distribution transverse to the fibres of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509034.png" />. Correspondingly, the linear differential forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509035.png" /> are a semi-basis of forms of the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509036.png" /> and form a completely-integrable subsystem of forms in the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509037.png" />. The fibres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509038.png" /> are integral manifolds of maximal dimension for the system of Pfaffian equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509039.png" /> (cf. [[Pfaffian equation|Pfaffian equation]]; [[Completely-integrable differential equation|Completely-integrable differential equation]]).
+
Let $  X _ {n} $
 +
be an $  n $-dimensional homogeneous space and let $  G $
 +
be the $  r $-dimensional [[Lie group|Lie group]] of its transformations ( $  G $
 +
acts from the left). Let $  X _ {n} = G/H $
 +
be a representation, where $  H \subset  G $
 +
is the [[Isotropy group|isotropy group]] (stationary group) of a certain point $  x _ {0} \in X _ {n} $;  
 +
let $  ( e _ {k} , e _  \alpha  ) $,
 +
$  k = 1, \dots, n $,
 +
$  \alpha = n+ 1, \dots, r $,  
 +
be a basis of left-invariant vector fields on $  G $
 +
such that the $  e _  \alpha  $
 +
restricted to $  H $
 +
also constitute a basis of left-invariant vector fields for the Lie subgroup $  H $.  
 +
The basis $  ( e _ {k} , e _  \alpha  ) $
 +
corresponds to a dual basis of left-invariant linear differential forms $  ( \theta  ^ {k} , \theta  ^  \alpha  ) $
 +
on the Lie group $  G $.  
 +
The canonical projection $  \pi : G \rightarrow X _ {n} $
 +
puts the points $  x \in X _ {n} $
 +
into correspondence with the left cosets $  \pi ( x) = H _ {x} \subset  G $
 +
of $  G $
 +
with respect to $  H = H _ {x _ {0}  } $,  
 +
and it introduces the structure of a principal $  H $-bundle with base $  X _ {n} $
 +
and structure group $  H $
 +
of dimension $  r- n $
 +
on the Lie group $  G $.  
 +
With this representation of $  G $,  
 +
the vector fields $  e _  \alpha  $
 +
constitute a basis of fundamental vector fields for the bundle $  \pi : G \rightarrow X _ {n} $,  
 +
while the vector fields $  e _ {k} $
 +
span a certain $  n $-distribution transverse to the fibres of $  \pi : G \rightarrow X _ {n} $.  
 +
Correspondingly, the linear differential forms $  \theta  ^ {k} $
 +
are a semi-basis of forms of the bundle $  \pi : G \rightarrow X _ {n} $
 +
and form a completely-integrable subsystem of forms in the system $  ( \theta  ^ {k} , \theta  ^  \alpha  ) $.  
 +
The fibres $  H _ {x} \subset  G $
 +
are integral manifolds of maximal dimension for the system of Pfaffian equations $  \theta  ^ {k} = 0 $ (cf. [[Pfaffian equation|Pfaffian equation]]; [[Completely-integrable differential equation|Completely-integrable differential equation]]).
 +
 
 +
A system of frames of reference in classical differential geometry (Euclidean, affine, projective, etc.) is a set of figures in  $  X _ {n} $
 +
that is in bijective correspondence with the set of transformations of  $  X _ {n} $ (or, which is the same, with the set of elements of the [[Fundamental group|fundamental group]]  $  G $
 +
of that space). Moreover, any frame of reference  $  R $
 +
from the given system can be obtained from some initial one  $  R _ {0} $
 +
by means of only one transformation:
 +
 
 +
$$
 +
L _ {g} :  X _ {n} \rightarrow X _ {n} ,\  R  =  L _ {g} ( R _ {0} ),\ \
 +
g \in G .
 +
$$
 +
 
 +
As the main role of the moving frame of reference  $  L _ {g} ( R _ {0} ) = R _ {g} $
 +
in relation to the fixed one  $  R _ {0} $
 +
is that it enables one to determine any transformation  $  L _ {g} $
 +
of the homogeneous space  $  X _ {n} $,
 +
one can identify the set of frames of reference  $  \{ R _ {g} \} $
 +
with the set of elements of the fundamental group  $  G $
 +
of the space, thus obtaining a notion of abstract frames of reference in any homogeneous space with given fundamental group  $  G $.
 +
 
 +
Let some smooth submanifold  $  M \subset  X _ {n} $
 +
of dimension  $  m $
 +
be given. Frames of order zero for  $  M $
 +
are elements of the restriction  $  G ( \pi , M) = G \mid  _ {M} \subset  G $
 +
of the bundle  $  \pi :  G \rightarrow X _ {n} $
 +
to  $  M $,
 +
as a new basis. This means that the principal bundle  $  G( \pi , M) \rightarrow M $
 +
is imbedded in  $  G $
 +
and is defined in it as the complete pre-image  $  \pi  ^ {- 1} ( M) \subset  G $.
 +
As the left-invariant forms  $  \theta  ^ {k} $
 +
and  $  \theta  ^  \alpha  $
 +
in the Lie group  $  G $
 +
satisfy the Maurer–Cartan equations
  
A system of frames of reference in classical differential geometry (Euclidean, affine, projective, etc.) is a set of figures in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509040.png" /> that is in bijective correspondence with the set of transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509041.png" /> (or, which is the same, with the set of elements of the [[Fundamental group|fundamental group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509042.png" /> of that space). Moreover, any frame of reference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509043.png" /> from the given system can be obtained from some initial one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509044.png" /> by means of only one transformation:
+
$$ \tag{1 }
 +
\left .
 +
\begin{array}{c}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509045.png" /></td> </tr></table>
+
d \theta  ^ {k}  =
 +
\frac{1}{2}
 +
C _ {lm}  ^ {k} \theta  ^ {l} \wedge
 +
\theta  ^ {m}
 +
+ C _ {l \alpha }  ^ {k} \theta  ^ {l} \wedge \theta  ^  \alpha  ,
 +
\\
  
As the main role of the moving frame of reference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509046.png" /> in relation to the fixed one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509047.png" /> is that it enables one to determine any transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509048.png" /> of the homogeneous space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509049.png" />, one can identify the set of frames of reference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509050.png" /> with the set of elements of the fundamental group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509051.png" /> of the space, thus obtaining a notion of abstract frames of reference in any homogeneous space with given fundamental group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509052.png" />.
+
d \theta  ^  \alpha  =
 +
\frac{1}{2}
 +
C _ {\beta \gamma }  ^  \alpha  \theta  ^  \beta
 +
\wedge \theta  ^  \gamma  +
 +
C _ {\beta k }  ^  \alpha  \theta  ^  \beta  \wedge \theta  ^ {k} +
  
Let some smooth submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509053.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509054.png" /> be given. Frames of order zero for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509055.png" /> are elements of the restriction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509056.png" /> of the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509057.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509058.png" />, as a new basis. This means that the principal bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509059.png" /> is imbedded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509060.png" /> and is defined in it as the complete pre-image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509061.png" />. As the left-invariant forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509062.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509063.png" /> in the Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509064.png" /> satisfy the Maurer–Cartan equations
+
\frac{1}{2}
 +
C _ {lm}  ^  \alpha  \theta  ^ {l} \wedge \theta  ^ {m} ,
 +
\\
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509065.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
k, l, m = 1, \dots, n; \  \alpha , \beta , \gamma = n+ 1, \dots, r,
 +
 +
\end{array}
 +
\right \}
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509067.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509068.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509069.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509070.png" /> are the structure constants of the Lie group, the restrictions of the forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509071.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509072.png" /> to the subbundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509073.png" />, i.e. the forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509074.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509075.png" />, will be subject to the same equations, but in addition between the forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509076.png" /> one has the linear relations
+
where $  C _ {lm}  ^ {k} $,  
 +
$  C _ {l _  \alpha  }  ^ {k} $,  
 +
$  C _ {\beta \gamma }  ^  \alpha  $,  
 +
$  C _ {\beta k }  ^  \alpha  $,  
 +
$  C _ {lm}  ^  \alpha  $
 +
are the structure constants of the Lie group, the restrictions of the forms $  \theta  ^ {k} $
 +
and $  \theta  ^  \alpha  $
 +
to the subbundle $  G( \pi , M) $,  
 +
i.e. the forms $  \omega  ^ {k} $
 +
and $  \omega  ^  \alpha  $,  
 +
will be subject to the same equations, but in addition between the forms $  \omega  ^ {k} $
 +
one has the linear relations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509077.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\omega  ^ {p}  = \Lambda _ {a}  ^ {p} \omega  ^ {a} ,\ \
 +
a = 1, \dots, m; \  p = m+ 1, \dots, n.
 +
$$
  
Here the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509078.png" /> are forms that remain, along with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509079.png" />, linearly independent in the principal bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509080.png" />, while the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509081.png" /> are functions also defined on the bundle of frames of order zero of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509082.png" />. The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509083.png" /> are coordinates in the tangent plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509084.png" /> of the submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509085.png" />, which depend on the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509086.png" /> and the frame
+
Here the $  \omega  ^ {a} $
 +
are forms that remain, along with $  \omega  ^  \alpha  $,  
 +
linearly independent in the principal bundle $  G( \pi , M) \rightarrow M $,  
 +
while the $  \Lambda _ {a}  ^ {p} $
 +
are functions also defined on the bundle of frames of order zero of $  G( \pi , M) \rightarrow M $.  
 +
The functions $  \Lambda _ {a}  ^ {p} $
 +
are coordinates in the tangent plane $  T _ {x} ( M) \subset  T _ {x} ( X _ {n} ) $
 +
of the submanifold $  M \subset  X _ {n} $,  
 +
which depend on the point $  x \in M $
 +
and the frame
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509087.png" /></td> </tr></table>
+
$$
 +
R  \in  \pi  ^ {- 1 }( x)  = H _ {x}  \subset  G( \pi , M).
 +
$$
  
The tangent planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509088.png" /> form a section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509089.png" /> of the Grassmann bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509090.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509091.png" />-planes passing through the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509092.png" />. The bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509093.png" /> is associated to the principal bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509094.png" />. The structure of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509095.png" /> is characterized by the equations
+
The tangent planes $  x \rightarrow T _ {x} ( M) $
 +
form a section $  f: M \rightarrow {\mathcal G} _ {m} ( M) $
 +
of the Grassmann bundle $  {\mathcal G} _ {m} ( M) \rightarrow M $
 +
of m $-planes passing through the points of $  M $.  
 +
The bundle $  {\mathcal G} _ {m} ( M) \rightarrow M $
 +
is associated to the principal bundle $  G( \pi , M) \rightarrow M $.  
 +
The structure of the functions $  \Lambda _ {a}  ^ {p} $
 +
is characterized by the equations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509096.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
d \Lambda _ {a}  ^ {p} + F _ {a \alpha }  ^ { p } ( \Lambda )
 +
\omega  ^  \alpha  =  \Lambda _ {ab}  ^ {p} \omega  ^ {b} ,
 +
$$
  
the explicit form of which can be obtained by exterior differentiation (cf. [[Exterior form|Exterior form]]) of (2) by means of (1) and subsequent application of Cartan's lemma (cf. [[Cartan lemma|Cartan lemma]]). The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509097.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509098.png" /> are the relative coordinates of the one-jet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m06509099.png" /> of the section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090100.png" /> in relation to the moving frame <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090101.png" /> for a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090102.png" />. The geometric object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090103.png" /> forms also a section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090104.png" /> of the corresponding bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090105.png" /> associated to the principal bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090106.png" />. Similarly one obtains the section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090107.png" /> with coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090108.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090109.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090110.png" /> of the generating geometric object, and also the subsequent extensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090111.png" />, which correspond to differential extensions of (3).
+
the explicit form of which can be obtained by exterior differentiation (cf. [[Exterior form|Exterior form]]) of (2) by means of (1) and subsequent application of Cartan's lemma (cf. [[Cartan lemma|Cartan lemma]]). The functions $  \Lambda _ {a}  ^ {p} $
 +
and $  \Lambda _ {ab}  ^ {p} $
 +
are the relative coordinates of the one-jet $  j _ {x}  ^ {1} f $
 +
of the section $  f $
 +
in relation to the moving frame $  R \in \pi  ^ {- 1} ( x) $
 +
for a point $  x \in M $.  
 +
The geometric object $  j _ {x}  ^ {1} f $
 +
forms also a section $  j  ^ {1} f: M \rightarrow {\mathcal G} _ {m}  ^ {1} ( M) $
 +
of the corresponding bundle $  {\mathcal G} _ {m}  ^ {1} ( M) \rightarrow ( M) $
 +
associated to the principal bundle $  G( \pi , M) \rightarrow M $.  
 +
Similarly one obtains the section $  j  ^ {2} f: M \rightarrow {\mathcal G} _ {m}  ^ {2} ( M) $
 +
with coordinates $  \Lambda _ {a}  ^ {p} $,  
 +
$  \Lambda _ {ab}  ^ {p} $,  
 +
$  \Lambda _ {abc}  ^ {p} $
 +
of the generating geometric object, and also the subsequent extensions $  j  ^ {3} f, \dots, j  ^ {q} f $,  
 +
which correspond to differential extensions of (3).
  
As long as the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090112.png" /> to which the section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090113.png" /> belongs is homogeneous, it is possible to perform a reduction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090114.png" /> of the principal bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090115.png" /> of frames to a certain subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090116.png" /> defined by Cartan by fixing the relative coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090117.png" /> for the geometric object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090118.png" />, which is independent of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090119.png" />. In this way one defines a partial canonization of the frame of reference. The frames <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090120.png" /> are called semi-canonical frames of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090121.png" /> for the given submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090122.png" />. If the subsequent continuation gives geometric objects whose isotropy group contains only the identity transformation, it is possible to fix only some of the coordinates of the geometric objects of the section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090123.png" /> which do not depend on the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090124.png" />, after which the other coordinates of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090125.png" /> of the geometric object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090126.png" /> depend only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090127.png" />. One thus gets a section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090128.png" /> of the bundle of frames of order zero of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090129.png" />. The frame <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090130.png" /> of this section is called the canonical frame of the submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090131.png" /> or the accompanying frame of this submanifold. The above process for the continuation of the equations (3) and the method selected for fixing the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090132.png" /> leads to the equations
+
As long as the bundle $  {\mathcal G} _ {m}  ^ {a} ( M) \rightarrow M $
 +
to which the section $  j  ^ {q} f( M) $
 +
belongs is homogeneous, it is possible to perform a reduction $  G  ^ {q} ( \pi , M) $
 +
of the principal bundle $  G( \pi , M) \rightarrow M $
 +
of frames to a certain subgroup $  \widetilde{H}  \subset  H $
 +
defined by Cartan by fixing the relative coordinates $  \Lambda _ {a}  ^ {p} , \Lambda _ {ab}  ^ {p}, \dots, \Lambda _ {a _ {1}  \dots a _ {q+ 1} }  ^ {p} $
 +
for the geometric object $  j _ {x}  ^ {q} f $,  
 +
which is independent of the point $  x \in M $.  
 +
In this way one defines a partial canonization of the frame of reference. The frames $  R \in G  ^ {q} ( \pi , M) $
 +
are called semi-canonical frames of order $  q+ 1 $
 +
for the given submanifold $  M \subset  X _ {n} $.  
 +
If the subsequent continuation gives geometric objects whose isotropy group contains only the identity transformation, it is possible to fix only some of the coordinates of the geometric objects of the section $  j  ^ {q+ 1} f $
 +
which do not depend on the point $  x $,  
 +
after which the other coordinates of $  \Lambda $
 +
of the geometric object $  j  ^ {q+ 1} f $
 +
depend only on $  x \in M $.  
 +
One thus gets a section $  s : M \rightarrow G( \pi , M) $
 +
of the bundle of frames of order zero of $  M $.  
 +
The frame $  R = s( x) $
 +
of this section is called the canonical frame of the submanifold $  M \subset  X _ {n} $
 +
or the accompanying frame of this submanifold. The above process for the continuation of the equations (3) and the method selected for fixing the function $  \Lambda $
 +
leads to the equations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090133.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$ \tag{4 }
 +
\omega  ^ {p}  = \Lambda _  \alpha  ^ {p} \omega  ^ {a} ,\ \
 +
\omega  ^  \alpha  = \Lambda _ {a}  ^  \alpha  \omega  ^ {a} ,
 +
$$
  
which connect the linear forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090134.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090135.png" /> in the section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090136.png" />. The field of the canonical frame is not constructed unambiguously, being dependent on arbitrarily fixing the relative coordinates of the geometric object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090137.png" />. The only important point is that some of the coefficients in (4) have constant numerical values (preferably the simplest ones), whereas the others form differential invariants for the submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090138.png" /> that define it up to transformations in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090139.png" />. The canonical frames for the section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090140.png" /> are analogues of a classical example, the accompanying Frénet frame (cf. [[Frénet trihedron|Frénet trihedron]]) for a curve in the Euclidean space, while equations (4) correspond to the Frénet equations (cf. [[Frénet formulas|Frénet formulas]]) for the curve. During the canonization of the frame of reference, complications may arise connected with the inhomogeneity of the bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090141.png" /> and the differences in type (in this sense) between the different submanifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090142.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090143.png" />, and even in individual parts of them. This is the basis for classifying the various types of points and various classes of submanifolds in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090144.png" />. On account of these features, the moving-frame method has played a fruitful part in research on submanifolds in various homogeneous spaces, and it has also indicated a way of developing modern methods of investigating very general differential-geometric structures on smooth manifolds.
+
which connect the linear forms $  \omega  ^ {k} $
 +
and $  \omega  ^  \alpha  $
 +
in the section $  s( M) $.  
 +
The field of the canonical frame is not constructed unambiguously, being dependent on arbitrarily fixing the relative coordinates of the geometric object $  j  ^ {q+ 1} f $.  
 +
The only important point is that some of the coefficients in (4) have constant numerical values (preferably the simplest ones), whereas the others form differential invariants for the submanifold $  M \subset  X _ {n} $
 +
that define it up to transformations in $  X _ {n} $.  
 +
The canonical frames for the section $  s( M) $
 +
are analogues of a classical example, the accompanying Frénet frame (cf. [[Frénet trihedron|Frénet trihedron]]) for a curve in the Euclidean space, while equations (4) correspond to the Frénet equations (cf. [[Frénet formulas|Frénet formulas]]) for the curve. During the canonization of the frame of reference, complications may arise connected with the inhomogeneity of the bundles $  {\mathcal G} _ {m}  ^ {q} ( M) $
 +
and the differences in type (in this sense) between the different submanifolds $  M $
 +
in $  X _ {n} $,  
 +
and even in individual parts of them. This is the basis for classifying the various types of points and various classes of submanifolds in $  X _ {n} $.  
 +
On account of these features, the moving-frame method has played a fruitful part in research on submanifolds in various homogeneous spaces, and it has also indicated a way of developing modern methods of investigating very general differential-geometric structures on smooth manifolds.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Cartan,  "La théorie des groupes finis et continus et la géométrie différentielle traitées par la méthode du repère mobile" , Gauthier-Villars  (1937)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Favard,  "Cours de géométrie différentielle locale" , Gauthier-Villars  (1957)</TD></TR><TR><TD valign="top">[3a]</TD> <TD valign="top">  H. Cartan,  "Differential forms" , Kershaw  (1983)  (Translated from French)</TD></TR><TR><TD valign="top">[3b]</TD> <TD valign="top">  H. Cartan,  "Calcul différentielle" , Hermann  (1967)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S.P. Finikov,  "Cartan's method of exterior forms in differential geometry" , '''1–3''' , Moscow-Leningrad  (1948)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Cartan,  "La théorie des groupes finis et continus et la géométrie différentielle traitées par la méthode du repère mobile" , Gauthier-Villars  (1937)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Favard,  "Cours de géométrie différentielle locale" , Gauthier-Villars  (1957)</TD></TR><TR><TD valign="top">[3a]</TD> <TD valign="top">  H. Cartan,  "Differential forms" , Kershaw  (1983)  (Translated from French)</TD></TR><TR><TD valign="top">[3b]</TD> <TD valign="top">  H. Cartan,  "Calcul différentielle" , Hermann  (1967)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S.P. Finikov,  "Cartan's method of exterior forms in differential geometry" , '''1–3''' , Moscow-Leningrad  (1948)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
In [[#References|[a3]]], Chapt. II, Sect. IV: La méthode du repère mobile, Cartan writes as follows.  "We shall resume the projective differentiable study of plane curves by attaching in an intrinsic way…a moving frame to the current point of the curve and study the properties of the curve by those of the displacements of the frame" .
 
In [[#References|[a3]]], Chapt. II, Sect. IV: La méthode du repère mobile, Cartan writes as follows.  "We shall resume the projective differentiable study of plane curves by attaching in an intrinsic way…a moving frame to the current point of the curve and study the properties of the curve by those of the displacements of the frame" .
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090145.png" /> be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090146.png" />-dimensional differentiable manifold and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090147.png" /> a point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090148.png" />. A [[Frame|frame]] at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090149.png" /> is a basis of the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090150.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090151.png" />. Given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090152.png" /> vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090153.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090154.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090155.png" /> are linearly independent for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090156.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090157.png" /> define a moving frame (repère mobile) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090158.png" />. Conversely, every moving frame <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090159.png" />, i.e. a section of the frame bundle (cf. [[Frame|Frame]]), determines such an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090160.png" />-tuple of vector fields. In Cartan's theory the basic idea is to express everything in terms of an arbitrary moving frame <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090161.png" /> and not just in terms of the  "natural"  frames <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090162.png" /> defined by a (local) system of coordinates. This has been a very fruitful idea, not at least because a moving frame may well exist on a region which cannot even be included in a coordinate system. For example, there is a rather obvious moving frame <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090163.png" /> on the whole torus. Similarly, there are the very useful moving frames defined in the main article above on the whole of the homogeneous spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090164.png" /> which are given by suitable left-invariant vector fields.
+
Let $  M $
 +
be $  n $-dimensional differentiable manifold and $  p $
 +
a point of $  M $.  
 +
A [[Frame|frame]] at $  p $
 +
is a basis of the tangent space $  T _ {p} M $
 +
at $  p \in M $.  
 +
Given $  n $
 +
vector fields $  X _ {1}, \dots, X _ {n} $
 +
on $  U \subset  M $
 +
such that $  X _ {1} ( q), \dots, X _ {n} ( q) $
 +
are linearly independent for each $  q \in U $,  
 +
the $  X _ {1} ( q), \dots, X _ {n} ( q) $
 +
define a moving frame (repère mobile) on $  U $.  
 +
Conversely, every moving frame $  p \mapsto F _ {p} \in ( T _ {p} M )  ^ {n} $,  
 +
i.e. a section of the frame bundle (cf. [[Frame|Frame]]), determines such an $  n $-tuple of vector fields. In Cartan's theory the basic idea is to express everything in terms of an arbitrary moving frame $  X _ {1}, \dots, X _ {n} $
 +
and not just in terms of the  "natural"  frames $  ( \partial  / {\partial  x _ {1} }, \dots, \partial  / {\partial  x _ {n} } ) $
 +
defined by a (local) system of coordinates. This has been a very fruitful idea, not at least because a moving frame may well exist on a region which cannot even be included in a coordinate system. For example, there is a rather obvious moving frame $  X _ {1} , X _ {2} $
 +
on the whole torus. Similarly, there are the very useful moving frames defined in the main article above on the whole of the homogeneous spaces $  G / H $
 +
which are given by suitable left-invariant vector fields.
  
On a [[Riemannian manifold|Riemannian manifold]] an orthonormal moving frame <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090165.png" /> is one for which the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090166.png" /> form an orthonormal basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090167.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065090/m065090168.png" />. An orthonormal moving frame can be obtained from an arbitrary one by Gram–Schmidt orthonormalization.
+
On a [[Riemannian manifold|Riemannian manifold]] an orthonormal moving frame $  X _ {1}, \dots, X _ {n} $
 +
is one for which the $  X _ {1} ( p), \dots, X _ {n} ( p) $
 +
form an orthonormal basis of $  T _ {p} M $
 +
for all $  p $.  
 +
An orthonormal moving frame can be obtained from an arbitrary one by Gram–Schmidt orthonormalization.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Jensen,  "Higher order contact of submanifolds of homogeneous spaces" , ''Lect. notes in math.'' , '''610''' , Springer  (1977)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Kobayashi,  K. Nomizu,  "Foundations of differential geometry" , '''1–2''' , Interscience  (1969)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E. Cartan,  "Théorie des espaces à connexion projective" , Gauthier-Villars  (1937)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Spivak,  "A comprehensive introduction to differential geometry" , '''II''' , Publish or Perish  (1970)  pp. Chapt. VII</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Jensen,  "Higher order contact of submanifolds of homogeneous spaces" , ''Lect. notes in math.'' , '''610''' , Springer  (1977)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Kobayashi,  K. Nomizu,  "Foundations of differential geometry" , '''1–2''' , Interscience  (1969)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E. Cartan,  "Théorie des espaces à connexion projective" , Gauthier-Villars  (1937)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Spivak,  "A comprehensive introduction to differential geometry" , '''II''' , Publish or Perish  (1970)  pp. Chapt. VII</TD></TR></table>

Latest revision as of 08:15, 18 August 2022


A method in differential geometry for the local examination of submanifolds of various homogeneous spaces, in which the starting point is to provide the submanifold itself and all its geometrical objects with the most general possible (mobile) frame (of reference). This method includes making the frame of reference canonical, namely assigning to each point in the submanifold a unique frame of reference in an invariant manner, in order to obtain differential invariants characterizing the submanifold apart from transformations imbedding it in the surrounding homogeneous space. This method was proposed in its most general form by E. Cartan [1], who gave various examples of its application. Subsequently, the method was widely used and developed (see Method of extensions and restrictions). The analytic basis of the method is constituted by the invariant linear differential forms of Lie groups and their structure equations, as well as by the theory of representations of Lie groups as transformation groups. In modern geometry, the basic concepts of the method have required refinement, and they have been formulated in terms of the theory of bundles.

Let $ X _ {n} $ be an $ n $-dimensional homogeneous space and let $ G $ be the $ r $-dimensional Lie group of its transformations ( $ G $ acts from the left). Let $ X _ {n} = G/H $ be a representation, where $ H \subset G $ is the isotropy group (stationary group) of a certain point $ x _ {0} \in X _ {n} $; let $ ( e _ {k} , e _ \alpha ) $, $ k = 1, \dots, n $, $ \alpha = n+ 1, \dots, r $, be a basis of left-invariant vector fields on $ G $ such that the $ e _ \alpha $ restricted to $ H $ also constitute a basis of left-invariant vector fields for the Lie subgroup $ H $. The basis $ ( e _ {k} , e _ \alpha ) $ corresponds to a dual basis of left-invariant linear differential forms $ ( \theta ^ {k} , \theta ^ \alpha ) $ on the Lie group $ G $. The canonical projection $ \pi : G \rightarrow X _ {n} $ puts the points $ x \in X _ {n} $ into correspondence with the left cosets $ \pi ( x) = H _ {x} \subset G $ of $ G $ with respect to $ H = H _ {x _ {0} } $, and it introduces the structure of a principal $ H $-bundle with base $ X _ {n} $ and structure group $ H $ of dimension $ r- n $ on the Lie group $ G $. With this representation of $ G $, the vector fields $ e _ \alpha $ constitute a basis of fundamental vector fields for the bundle $ \pi : G \rightarrow X _ {n} $, while the vector fields $ e _ {k} $ span a certain $ n $-distribution transverse to the fibres of $ \pi : G \rightarrow X _ {n} $. Correspondingly, the linear differential forms $ \theta ^ {k} $ are a semi-basis of forms of the bundle $ \pi : G \rightarrow X _ {n} $ and form a completely-integrable subsystem of forms in the system $ ( \theta ^ {k} , \theta ^ \alpha ) $. The fibres $ H _ {x} \subset G $ are integral manifolds of maximal dimension for the system of Pfaffian equations $ \theta ^ {k} = 0 $ (cf. Pfaffian equation; Completely-integrable differential equation).

A system of frames of reference in classical differential geometry (Euclidean, affine, projective, etc.) is a set of figures in $ X _ {n} $ that is in bijective correspondence with the set of transformations of $ X _ {n} $ (or, which is the same, with the set of elements of the fundamental group $ G $ of that space). Moreover, any frame of reference $ R $ from the given system can be obtained from some initial one $ R _ {0} $ by means of only one transformation:

$$ L _ {g} : X _ {n} \rightarrow X _ {n} ,\ R = L _ {g} ( R _ {0} ),\ \ g \in G . $$

As the main role of the moving frame of reference $ L _ {g} ( R _ {0} ) = R _ {g} $ in relation to the fixed one $ R _ {0} $ is that it enables one to determine any transformation $ L _ {g} $ of the homogeneous space $ X _ {n} $, one can identify the set of frames of reference $ \{ R _ {g} \} $ with the set of elements of the fundamental group $ G $ of the space, thus obtaining a notion of abstract frames of reference in any homogeneous space with given fundamental group $ G $.

Let some smooth submanifold $ M \subset X _ {n} $ of dimension $ m $ be given. Frames of order zero for $ M $ are elements of the restriction $ G ( \pi , M) = G \mid _ {M} \subset G $ of the bundle $ \pi : G \rightarrow X _ {n} $ to $ M $, as a new basis. This means that the principal bundle $ G( \pi , M) \rightarrow M $ is imbedded in $ G $ and is defined in it as the complete pre-image $ \pi ^ {- 1} ( M) \subset G $. As the left-invariant forms $ \theta ^ {k} $ and $ \theta ^ \alpha $ in the Lie group $ G $ satisfy the Maurer–Cartan equations

$$ \tag{1 } \left . \begin{array}{c} d \theta ^ {k} = \frac{1}{2} C _ {lm} ^ {k} \theta ^ {l} \wedge \theta ^ {m} + C _ {l \alpha } ^ {k} \theta ^ {l} \wedge \theta ^ \alpha , \\ d \theta ^ \alpha = \frac{1}{2} C _ {\beta \gamma } ^ \alpha \theta ^ \beta \wedge \theta ^ \gamma + C _ {\beta k } ^ \alpha \theta ^ \beta \wedge \theta ^ {k} + \frac{1}{2} C _ {lm} ^ \alpha \theta ^ {l} \wedge \theta ^ {m} , \\ k, l, m = 1, \dots, n; \ \alpha , \beta , \gamma = n+ 1, \dots, r, \end{array} \right \} $$

where $ C _ {lm} ^ {k} $, $ C _ {l _ \alpha } ^ {k} $, $ C _ {\beta \gamma } ^ \alpha $, $ C _ {\beta k } ^ \alpha $, $ C _ {lm} ^ \alpha $ are the structure constants of the Lie group, the restrictions of the forms $ \theta ^ {k} $ and $ \theta ^ \alpha $ to the subbundle $ G( \pi , M) $, i.e. the forms $ \omega ^ {k} $ and $ \omega ^ \alpha $, will be subject to the same equations, but in addition between the forms $ \omega ^ {k} $ one has the linear relations

$$ \tag{2 } \omega ^ {p} = \Lambda _ {a} ^ {p} \omega ^ {a} ,\ \ a = 1, \dots, m; \ p = m+ 1, \dots, n. $$

Here the $ \omega ^ {a} $ are forms that remain, along with $ \omega ^ \alpha $, linearly independent in the principal bundle $ G( \pi , M) \rightarrow M $, while the $ \Lambda _ {a} ^ {p} $ are functions also defined on the bundle of frames of order zero of $ G( \pi , M) \rightarrow M $. The functions $ \Lambda _ {a} ^ {p} $ are coordinates in the tangent plane $ T _ {x} ( M) \subset T _ {x} ( X _ {n} ) $ of the submanifold $ M \subset X _ {n} $, which depend on the point $ x \in M $ and the frame

$$ R \in \pi ^ {- 1 }( x) = H _ {x} \subset G( \pi , M). $$

The tangent planes $ x \rightarrow T _ {x} ( M) $ form a section $ f: M \rightarrow {\mathcal G} _ {m} ( M) $ of the Grassmann bundle $ {\mathcal G} _ {m} ( M) \rightarrow M $ of $ m $-planes passing through the points of $ M $. The bundle $ {\mathcal G} _ {m} ( M) \rightarrow M $ is associated to the principal bundle $ G( \pi , M) \rightarrow M $. The structure of the functions $ \Lambda _ {a} ^ {p} $ is characterized by the equations

$$ \tag{3 } d \Lambda _ {a} ^ {p} + F _ {a \alpha } ^ { p } ( \Lambda ) \omega ^ \alpha = \Lambda _ {ab} ^ {p} \omega ^ {b} , $$

the explicit form of which can be obtained by exterior differentiation (cf. Exterior form) of (2) by means of (1) and subsequent application of Cartan's lemma (cf. Cartan lemma). The functions $ \Lambda _ {a} ^ {p} $ and $ \Lambda _ {ab} ^ {p} $ are the relative coordinates of the one-jet $ j _ {x} ^ {1} f $ of the section $ f $ in relation to the moving frame $ R \in \pi ^ {- 1} ( x) $ for a point $ x \in M $. The geometric object $ j _ {x} ^ {1} f $ forms also a section $ j ^ {1} f: M \rightarrow {\mathcal G} _ {m} ^ {1} ( M) $ of the corresponding bundle $ {\mathcal G} _ {m} ^ {1} ( M) \rightarrow ( M) $ associated to the principal bundle $ G( \pi , M) \rightarrow M $. Similarly one obtains the section $ j ^ {2} f: M \rightarrow {\mathcal G} _ {m} ^ {2} ( M) $ with coordinates $ \Lambda _ {a} ^ {p} $, $ \Lambda _ {ab} ^ {p} $, $ \Lambda _ {abc} ^ {p} $ of the generating geometric object, and also the subsequent extensions $ j ^ {3} f, \dots, j ^ {q} f $, which correspond to differential extensions of (3).

As long as the bundle $ {\mathcal G} _ {m} ^ {a} ( M) \rightarrow M $ to which the section $ j ^ {q} f( M) $ belongs is homogeneous, it is possible to perform a reduction $ G ^ {q} ( \pi , M) $ of the principal bundle $ G( \pi , M) \rightarrow M $ of frames to a certain subgroup $ \widetilde{H} \subset H $ defined by Cartan by fixing the relative coordinates $ \Lambda _ {a} ^ {p} , \Lambda _ {ab} ^ {p}, \dots, \Lambda _ {a _ {1} \dots a _ {q+ 1} } ^ {p} $ for the geometric object $ j _ {x} ^ {q} f $, which is independent of the point $ x \in M $. In this way one defines a partial canonization of the frame of reference. The frames $ R \in G ^ {q} ( \pi , M) $ are called semi-canonical frames of order $ q+ 1 $ for the given submanifold $ M \subset X _ {n} $. If the subsequent continuation gives geometric objects whose isotropy group contains only the identity transformation, it is possible to fix only some of the coordinates of the geometric objects of the section $ j ^ {q+ 1} f $ which do not depend on the point $ x $, after which the other coordinates of $ \Lambda $ of the geometric object $ j ^ {q+ 1} f $ depend only on $ x \in M $. One thus gets a section $ s : M \rightarrow G( \pi , M) $ of the bundle of frames of order zero of $ M $. The frame $ R = s( x) $ of this section is called the canonical frame of the submanifold $ M \subset X _ {n} $ or the accompanying frame of this submanifold. The above process for the continuation of the equations (3) and the method selected for fixing the function $ \Lambda $ leads to the equations

$$ \tag{4 } \omega ^ {p} = \Lambda _ \alpha ^ {p} \omega ^ {a} ,\ \ \omega ^ \alpha = \Lambda _ {a} ^ \alpha \omega ^ {a} , $$

which connect the linear forms $ \omega ^ {k} $ and $ \omega ^ \alpha $ in the section $ s( M) $. The field of the canonical frame is not constructed unambiguously, being dependent on arbitrarily fixing the relative coordinates of the geometric object $ j ^ {q+ 1} f $. The only important point is that some of the coefficients in (4) have constant numerical values (preferably the simplest ones), whereas the others form differential invariants for the submanifold $ M \subset X _ {n} $ that define it up to transformations in $ X _ {n} $. The canonical frames for the section $ s( M) $ are analogues of a classical example, the accompanying Frénet frame (cf. Frénet trihedron) for a curve in the Euclidean space, while equations (4) correspond to the Frénet equations (cf. Frénet formulas) for the curve. During the canonization of the frame of reference, complications may arise connected with the inhomogeneity of the bundles $ {\mathcal G} _ {m} ^ {q} ( M) $ and the differences in type (in this sense) between the different submanifolds $ M $ in $ X _ {n} $, and even in individual parts of them. This is the basis for classifying the various types of points and various classes of submanifolds in $ X _ {n} $. On account of these features, the moving-frame method has played a fruitful part in research on submanifolds in various homogeneous spaces, and it has also indicated a way of developing modern methods of investigating very general differential-geometric structures on smooth manifolds.

References

[1] E. Cartan, "La théorie des groupes finis et continus et la géométrie différentielle traitées par la méthode du repère mobile" , Gauthier-Villars (1937)
[2] J. Favard, "Cours de géométrie différentielle locale" , Gauthier-Villars (1957)
[3a] H. Cartan, "Differential forms" , Kershaw (1983) (Translated from French)
[3b] H. Cartan, "Calcul différentielle" , Hermann (1967)
[4] S.P. Finikov, "Cartan's method of exterior forms in differential geometry" , 1–3 , Moscow-Leningrad (1948) (In Russian)

Comments

In [a3], Chapt. II, Sect. IV: La méthode du repère mobile, Cartan writes as follows. "We shall resume the projective differentiable study of plane curves by attaching in an intrinsic way…a moving frame to the current point of the curve and study the properties of the curve by those of the displacements of the frame" .

Let $ M $ be $ n $-dimensional differentiable manifold and $ p $ a point of $ M $. A frame at $ p $ is a basis of the tangent space $ T _ {p} M $ at $ p \in M $. Given $ n $ vector fields $ X _ {1}, \dots, X _ {n} $ on $ U \subset M $ such that $ X _ {1} ( q), \dots, X _ {n} ( q) $ are linearly independent for each $ q \in U $, the $ X _ {1} ( q), \dots, X _ {n} ( q) $ define a moving frame (repère mobile) on $ U $. Conversely, every moving frame $ p \mapsto F _ {p} \in ( T _ {p} M ) ^ {n} $, i.e. a section of the frame bundle (cf. Frame), determines such an $ n $-tuple of vector fields. In Cartan's theory the basic idea is to express everything in terms of an arbitrary moving frame $ X _ {1}, \dots, X _ {n} $ and not just in terms of the "natural" frames $ ( \partial / {\partial x _ {1} }, \dots, \partial / {\partial x _ {n} } ) $ defined by a (local) system of coordinates. This has been a very fruitful idea, not at least because a moving frame may well exist on a region which cannot even be included in a coordinate system. For example, there is a rather obvious moving frame $ X _ {1} , X _ {2} $ on the whole torus. Similarly, there are the very useful moving frames defined in the main article above on the whole of the homogeneous spaces $ G / H $ which are given by suitable left-invariant vector fields.

On a Riemannian manifold an orthonormal moving frame $ X _ {1}, \dots, X _ {n} $ is one for which the $ X _ {1} ( p), \dots, X _ {n} ( p) $ form an orthonormal basis of $ T _ {p} M $ for all $ p $. An orthonormal moving frame can be obtained from an arbitrary one by Gram–Schmidt orthonormalization.

References

[a1] G. Jensen, "Higher order contact of submanifolds of homogeneous spaces" , Lect. notes in math. , 610 , Springer (1977)
[a2] S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1–2 , Interscience (1969)
[a3] E. Cartan, "Théorie des espaces à connexion projective" , Gauthier-Villars (1937)
[a4] M. Spivak, "A comprehensive introduction to differential geometry" , II , Publish or Perish (1970) pp. Chapt. VII
How to Cite This Entry:
Moving-frame method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Moving-frame_method&oldid=17828
This article was adapted from an original article by E.L. Evtushik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article