Moving-frame method

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