Cartan method of exterior forms

A differential-algebraic method of studying systems of differential equations and manifolds with various structures. The algebraic basis of the method is the Grassmann algebra. Let $ V $ be a $ 2 ^ {n} $- dimensional vector space over an arbitrary field $ K $ with basis vectors $ e ^ {0} , e ^ {i} , e ^ {ij} \dots e ^ {1 \dots n } $, $ i \leq j < k \leq n $. In addition to the basis vectors, one defines for any natural number $ q $ the vectors $ e ^ {i _ {1} \dots i _ {q} } $, $ i _ {1} \dots i _ {q} = 1 \dots n $, according to the following rule: If at least two of the natural numbers $ i _ {1} \dots i _ {q} $ are identical, then $ e ^ {i _ {1} \dots i _ {q} } = 0 $; if all the $ i _ {1} \dots i _ {q} $ are distinct and the numbers $ j _ {1} < \dots < j _ {q} $ are a permutation of $ i _ {1} \dots i _ {q} $, then $ e ^ {i _ {1} \dots i _ {q} } = e ^ {j _ {1} \dots j _ {q} } $ if the permutation $ i _ {k} \rightarrow j _ {k} $, $ k = 1 \dots q $, is even, and $ e ^ {i _ {1} \dots i _ {q} } = - e ^ {j _ {1} \dots j _ {q} } $ if this permutation is odd. In the vector space $ V $ the exterior product is defined: $ e ^ {i _ {1} \dots i _ {p} } \wedge e ^ {k _ {1} \dots k _ {q} } = e ^ {i _ {1} \dots i _ {p} k _ {1} \dots k _ {q} } $; in addition, the usual laws for a hypercomplex system (i.e. an associative algebra) are required to hold. The algebra of dimension $ 2 ^ {n} $ over $ K $ so constructed is called the Grassmann algebra. A vector of the form

$$ \lambda e ^ {i _ {1} \dots i _ {p} } = \ \lambda e ^ {i _ {1} } \wedge \dots \wedge e ^ {i _ {p} } $$

is called a monomial of degree $ p $, $ \lambda \in K $. A sum of monomials of the same degree $ p > 1 $ is called an exterior form of degree $ p $; a sum of monomials of the first degree is called a linear form. The elements of the field $ K $ are, by definition, forms of degree zero. The vectors $ e ^ {i} $ generate the Grassmann algebra and so do any $ n $ linearly independent combinations of them

$$ \omega ^ {j} = \ a _ {i} ^ {j} e ^ {i} ,\ \ \mathop{\rm det} ( a _ {i} ^ {j} ) \neq 0,\ \ a _ {i} ^ {j} \in K. $$

Here and in what follows, identical indices occurring in pairs, once up and once down, are to be summed over the appropriate range.

By the first-order algebraic derivative of the exterior form

$$ \Omega _ {p} = \ a _ {i _ {1} \dots i _ {p} } e ^ {i _ {1} } \wedge \dots \wedge e ^ {i _ {p} } $$

of degree $ p $ with respect to the symbol $ e ^ {i} $ is meant the form $ \Omega _ {p - 1 } = {\partial \Omega _ {p} } / {\partial e ^ {i} } $ of degree $ p - 1 $, obtained from $ \Omega _ {p} $ by replacing by zero all monomials not containing the symbol $ e ^ {i} $, while for each of the remaining monomials the symbol $ e ^ {i} $ is first of all brought to the leftmost position with a change of sign for each successive shift to the left, and then replaced by one. The set of all $ (p - 1) $- th order non-zero algebraic derivatives of the form $ \Omega _ {p} $ is called the associated system of linear forms of $ \Omega _ {p} $. The rank of the exterior form $ \Omega _ {p} $ is the rank of its associated system. It is equal to the minimum number of linear forms in terms of which $ \Omega _ {p} $ can be expressed using the exterior product operation. For the study of a system of differential equations in $ \mathbf R ^ {n} $, the differential Grassmann algebra is used, where $ K $ is taken to be the ring of analytic functions in $ n $ real variables $ x ^ {i} $ defined in some domain of $ \mathbf R ^ {n} $, and the vectors $ e ^ {i} $ are denoted by $ dx ^ {i} $. Its linear forms are called $ 1 $- forms or Pfaffian forms, where the symbols $ dx ^ {i} $ are the differentials of the variables $ x ^ {i} $. The exterior forms of degree $ p > 1 $ are called $ p $- forms or exterior differential forms of degree $ p $. By the exterior differential of the $ p $- form

$$ \Omega _ {p} = \ a _ {i _ {1} \dots i _ {p} } dx ^ {i _ {1} } \wedge \dots \wedge dx ^ {i _ {p} } $$

is meant the $ (p + 1) $- form

$$ D \Omega _ {p} = \ da _ {i _ {1} \dots i _ {p} } \wedge dx ^ {i _ {1} } \wedge \dots \wedge dx ^ {i _ {p} } . $$

The exterior differential has the following properties:

$$ D ( \Omega _ {p} \pm \Omega _ {p} ^ {*} ) = \ D \Omega _ {p} \pm D \Omega _ {p} ^ {*} , $$

$$ D ( \Omega _ {p} \wedge \Omega _ {q} ) = D \Omega _ {p} \wedge \Omega _ {q} + (-1) ^ {p} \Omega _ {p} \wedge D \Omega _ {q} , $$

$$ D (D \Omega _ {p} ) \equiv 0, $$

where $ \Omega _ {p} $, $ \Omega _ {p} ^ {*} $ are arbitrary $ p $- forms and $ \Omega _ {q} $ is an arbitrary $ q $- form.

A Pfaffian form $ \omega = a _ {i} ds ^ {i} $ is locally the total differential of some function $ f $ if and only if its exterior differential vanishes. Let

$$ \tag{1 } \theta ^ \alpha \equiv \ b _ {a} ^ \alpha (x ^ {b} , z ^ {p} ) \ dx ^ \alpha + c _ \xi ^ \alpha (x ^ {b} , z ^ {p} ) \ dz ^ \xi - dz ^ \alpha = 0, $$

$$ \alpha = 1 \dots s; \ a, b = 1 \dots m; \ \xi = s + 1 \dots r; $$

$$ p = 1 \dots r, $$

be an arbitrary system of linearly independent Pfaffian equations in $ m $ independent variables $ x ^ {a} $ and $ r $ unknown functions $ z ^ {p} $. The system $ D \theta ^ \alpha = 0 $ is called the closure of the system (1). The closure is called pure closure (denoted by $ \overline{ {D \theta ^ \alpha }}\; = 0 $) if the original system (1) is algebraically accounted for in it, that is, if the quantities $ dz ^ \alpha $ in (1) are substituted into the quadratic forms $ D \theta ^ \alpha $. The system $ \theta ^ \alpha = 0 $, $ D \theta ^ \alpha = 0 $, or the system $ \theta ^ \alpha = 0 $, $ \overline{ {D \theta ^ \alpha }}\; = 0 $ equivalent to it, is called a closed system. The system (1) is completely integrable if and only if $ \overline{ {D \theta ^ \alpha }}\; = 0 $. Equating to zero the algebraic derivatives of $ \overline{ {D \theta ^ \alpha }}\; $ with respect to $ dx ^ {a} $ and $ dz ^ \xi $, $ a = 1 \dots m $; $ \xi = s + 1 \dots r $, and adjoining the Pfaffian equations to the original system (1), one obtains a completely integrable system of equations, called the characteristic system of (1). The set of its independent first integrals forms the smallest collection of variables in terms of which all equations of the system (1) can be expressed. Let $ m _ {\xi , h } ^ \alpha $ be the result of substituting for $ dx ^ {a} , dz ^ \xi $ in the algebraic derivative $ { {\partial D \theta ^ \alpha } / {\partial dz ^ \xi } } bar $ the arbitrary variables $ x _ {h} ^ {a} , z _ {h} ^ \xi $, $ h = 1 \dots m - 1 $. Associated with the system (1) is the sequence of matrices

$$ M _ {h} = \ \left ( \begin{array}{c} m _ {\xi , 1 } ^ \alpha \\ \dots \\ m _ {\xi , h } ^ \alpha \\ \end{array} \right ) . $$

The numbers

$$ s _ {1} = \mathop{\rm rank} M _ {1} , $$

$$ s _ {2} = \mathop{\rm rank} M _ {2} - \mathop{\rm rank} M _ {1} , $$

$$ {\dots \dots \dots \dots \dots } $$

$$ s _ {m - 1 } = \mathop{\rm rank} M _ {m - 1 } - \mathop{\rm rank} M _ {m - 2 } , $$

$$ s _ {m} = r - s - \mathop{\rm rank} M _ {m - 1 } $$

are called the characteristics and the number

$$ Q = s _ {1} + 2s _ {2} + \dots + ms _ {m} $$

is called the Cartan number of the system (1). By adjoining to the closed system $ \theta ^ \alpha = 0 $, $ \overline{ {D \theta ^ \alpha }}\; = 0 $ the equations $ dz ^ \xi = b _ {a} ^ \xi dx ^ {a} $, where the $ b _ {a} ^ \xi $ are new unknown functions, one obtains the first prolongation of the system (1). Let $ N $ be the number of functionally independent functions among the $ b _ {a} ^ \xi $. Then always $ N \leq Q $. If $ N = Q $, then the system (1) is in involution and its general solution depends on $ s _ {m} $ arbitrary functions in $ m $ arguments, $ s _ {m - 1 } $ functions in $ m - 1 $ arguments, etc., $ s _ {1} $ functions in one argument and $ s $ arbitrary constants. If, on the other hand, $ N < Q $, then (1) needs to be prolongated; after a finite number of prolongations one obtains either a system in involution or an inconsistent system.

Suppose, for example, that the system is

$$ dz _ {1} = \ u dx + x ^ {2} dy,\ \ dz _ {2} = \ u dy + y ^ {2} dx, $$

with independent variables $ x, y $ and unknown functions $ u , z _ {1} , z _ {2} $( $ s = 2= m $, $ r = 3 $). Its pure closure has the form:

$$ du \wedge dx + 2x \ dx \wedge dy = 0,\ \ du \wedge dy + 2y \ dy \wedge dx = 0. $$

For this system:

$$ M _ {1} = \ \left ( \begin{array}{c} X _ {1} \\ Y _ {1} \end{array} \right ) ,\ \ s _ {1} = \mathop{\rm rank} M _ {1} = 1,\ \ s _ {2} = 0,\ \ Q = 1,\ N = 0. $$

The system is not in involution. The prolongated system

$$ dz _ {1} = \ u dx + x ^ {2} dy,\ \ dz _ {2} = \ u dy + y ^ {2} dx,\ \ du = 2 (y dx + x dy) $$

is completely integrable and its general solution has the form

$$ u = 2xy + c _ {1} ,\ \ z _ {1} = \ x (xy + c _ {1} ) + c _ {2} ,\ \ z _ {2} = \ y (xy + c _ {1} ) + c _ {3} , $$

where $ c _ {1} , c _ {2} , c _ {3} $ are arbitrary constants.

Application of the Cartan method of exterior forms appreciably simplifies the statements and proofs of many theorems in mathematics and theoretical mechanics. For example, the Ostrogradski theorem is given by the formula

$$ \oint _ \Gamma \Omega = \ \int\limits _ { M } D \Omega , $$

where $ M $ is an analytic oriented $ (M + 1) $- dimensional manifold, $ \Gamma $ is its $ m $- dimensional smooth boundary, $ \Omega $ is an $ m $- form, and $ D \Omega $ is its exterior differential. The formula for the change of variables in a multiple integral

$$ J = {\int\limits \dots \int\limits } _ { D } f (x ^ {1} \dots x ^ {n} ) \ dx ^ {1} \wedge \dots \wedge dx ^ {n} $$

under a mapping $ p: \Delta \rightarrow D $, defined by the formulas $ x ^ {i} = \phi ^ {i} (u ^ {1} \dots u ^ {n} ) $, where $ D, \Delta \subset \mathbf R ^ {n} $, is obtained by the direct change of the variables $ x ^ {i} $ and their differentials $ dx ^ {i} = ( {\partial \phi ^ {i} } / {\partial u ^ {j} } ) du ^ {j} $. Since

$$ dx ^ {1} \wedge \dots \wedge dx ^ {n} = \ \frac{\partial ( \phi ^ {1} \dots \phi ^ {n} ) }{\partial (u ^ {1} \dots u ^ {n} ) } \ du ^ {1} \wedge \dots \wedge du ^ {n} , $$

it follows that

$$ J = {\int\limits \dots \int\limits } _ \Delta \frac{\partial ( \phi ^ {1} \dots \phi ^ {n} ) }{\partial (u ^ {1} \dots u ^ {n} ) } \ du ^ {1} \wedge \dots \wedge du ^ {n} . $$

Cartan's method of exterior forms is extensively used in the study of manifolds with various structures. Let $ M $ be a differentiable manifold of class $ C ^ \infty $, let $ F = C ^ \infty (M ) $ be the set of differentiable functions on $ M $, let $ D ^ {1} $ be the set of all the vector fields on $ M $, and let $ \mathfrak A _ {s} $ be the set of skew-symmetric $ F $- multilinear mappings on the module $ D ^ {1} \times \dots \times D ^ {1} $( $ s $ copies, where $ s \geq 1 $ is a natural number).

Let $ \mathfrak A _ {0} = F $ and denote by $ \mathfrak A $ the direct sum of the $ \mathfrak A _ {s} $:

$$ \mathfrak A = \ \sum _ {s = 0 } ^ \infty \mathfrak A _ {s} . $$

The elements of the module $ \mathfrak A $ are called exterior differential forms on $ M $; the elements of $ \mathfrak A _ {s} $ are called $ s $- forms. Let

$$ f, g \in C ^ \infty (M ); \ \ \theta \in \mathfrak A _ {r} ,\ \ \Omega \in \mathfrak A _ {s} ,\ \ X _ {i} \in D ^ {1} . $$

Then their exterior product $ \wedge $ is defined by the formulas:

$$ f \wedge g = fg,\ \ (f \wedge \theta ) (X _ {1} \dots X _ {r} ) = \ f \theta (X _ {1} \dots X _ {r} ), $$

$$ ( \Omega \wedge g) (X _ {1} \dots X _ {s} ) = g \Omega (X _ {1} \dots X _ {s} ), $$

$$ ( \theta \wedge \Omega ) (X _ {1} \dots X _ {r + s } ) = $$

$$ = \ { \frac{1}{(r + s)! } } \sum _ {\sigma \in S _ {r + s } } \epsilon ( \sigma ) \theta (X _ {\sigma (1) } \dots X _ {\sigma (r) } ) \times $$

$$ \times \Omega (X _ {\sigma (r + 1) } \dots X _ {\sigma (r + s) } ), $$

where $ S _ {r + s } $ is the group of permutations of the set $ 1 \dots r + s $, and $ \epsilon ( \sigma ) = 1 $ or $ -1 $ depending on whether the permutation $ \sigma $ is even or odd. The module $ \mathfrak A $ of skew-symmetric $ F $- multilinear functions with the exterior product is called the Grassmann algebra over the manifold $ M $. If $ M $ is $ \mathbf R ^ {n} $, then one obtains the differential Grassmann algebra considered earlier. By exterior differentiation one means the $ \mathbf R $- linear mapping $ D: \mathfrak A \rightarrow \mathfrak A $ with the following properties: $ D \mathfrak A _ {s} \subset \mathfrak A _ {s + 1 } $ for every $ s \geq 0 $; if $ f \in \mathfrak A _ {0} = C ^ \infty (M) $, then $ Df $ is the $ 1 $- form defined by the formula $ Df (X) = X (f) $, where $ X \in D ^ {1} $; $ D \cdot D = 0 $, $ D ( \theta \wedge \Omega ) = D \theta \wedge \Omega + (-1) ^ {r} \theta \wedge D \Omega $, if $ \theta \in \mathfrak A _ {r} $, $ \Omega \in \mathfrak A $. Suppose, for instance, that $ M $ is a manifold with a given affine connection. An affine connection on a manifold $ M $ is a rule $ \nabla $ which associates with each $ X \in D ^ {1} $ a linear mapping $ \nabla _ {X} $ of the vector space $ D ^ {1} $ into itself, satisfying the following two properties:

$$ \nabla _ {f X + gY } = \ f \nabla _ {X} + g \nabla _ {Y} ; \ \ \nabla _ {X} (f Y) = \ f \nabla _ {X} + (Xf ) Y $$

for $ f, g \in C ^ \infty (M ) $, $ X, Y \in D ^ {1} $. The operator $ \nabla _ {X} $ is called the covariant derivative with respect to $ X $. Let $ \Phi $ be a diffeomorphism of $ M $, and $ \nabla $ an affine connection on $ M $. The formula

$$ \nabla _ {X} ^ \prime (Y) = \ ( \nabla _ {X} \Phi (Y ^ \Phi )) ^ {\Phi - 1 } , $$

where $ X, Y \in D ^ {1} $, defines a new affine connection on $ M $. One says that $ \nabla $ is invariant with respect to $ \Phi $ if $ \nabla ^ \prime = \nabla $. In this case $ \Phi $ is called an affine transformation of $ M $. Let

$$ [X, Y] = XY - YX, $$

$$ T (X, Y) = \nabla _ {X} (Y) - \nabla _ {Y} (X) - [X, Y], $$

$$ R (X, Y) = \nabla _ {X} \nabla _ {Y} - \nabla _ {Y} \nabla _ {X} - \nabla _ {[X, Y] } , $$

for all $ X, Y \in D ^ {1} $, and let $ D _ {1} $ be the module dual to the $ F $- module $ D ^ {1} $. The $ F $- multilinear mapping $ ( \omega , X, Y) \rightarrow \omega (T (X, Y)) $, where $ \omega \in D _ {1} $ is a Pfaffian form, is called the torsion tensor field, and is denoted by $ T $; the $ F $- multilinear mapping $ ( \omega , Z, X, Y) \rightarrow \omega (R (X, Y) \cdot Z) $ is called the curvature tensor field, and is denoted by $ R $. Let $ p \in M $ and let $ X _ {1} \dots X _ {n} $ be a basis for the vector fields in some neighbourhood $ U _ {p} $ of the point $ p $. The functions $ \Gamma _ {IJ} ^ {K} $, $ T _ {IJ} ^ {K} $, $ R _ {IJ} ^ {K} $ are defined on $ U _ {p} $ by the formulas

$$ \nabla _ {X _ {J} } (X _ {J} ) = \ \Gamma _ {IJ} ^ {K} X _ {K} ,\ \ T (X _ {I} , X _ {J} ) = \ T _ {IJ} ^ {K} X _ {K} , $$

$$ R (X _ {I} , X _ {J} ) \cdot X _ {L} = R _ {LIJ} ^ {K} X _ {K} ,\ I, J, K = 1 \dots n. $$

For the $ 1 $- forms $ \omega ^ {I} $, $ \omega _ {J} ^ {K} $ defined on $ U _ {p} $ by the formulas

$$ \omega ^ {I} (X _ {J} ) = \ \delta _ {J} ^ {I} ,\ \ \omega _ {I} ^ {J} = \ \Gamma _ {KI} ^ {J} \omega ^ {K} , $$

the following structural equations of Cartan hold:

$$ D \omega ^ {J} = \ \omega ^ {K} \wedge \omega _ {K} ^ {J} + { \frac{1}{2} } T _ {JK} ^ {I} \omega ^ {J} \wedge \omega ^ {K} , $$

$$ D \omega _ {I} ^ {J} = \omega _ {I} ^ {K} \wedge \omega _ {K} ^ {J} + { \frac{1}{2} } R _ {IKL} ^ {J} \omega ^ {K} \wedge \omega ^ {L} . $$

The system of Pfaffian equations

$$ \omega ^ {a} = \ \lambda _ {i} ^ {a} \omega ^ {i} ,\ \ i, j, k = 1 \dots m; \ a, b = m + 1 \dots n, $$

defines an $ m $- dimensional submanifold $ \mathfrak M _ {m} \subset M $. Extending this system by means of the Cartan structural equations, one obtains a sequence of fundamental geometric objects of the submanifold

$$ \mathfrak M _ {m} :\ \{ \lambda _ {i} ^ {a} \} ,\ \ \{ \lambda _ {i} ^ {a} ,\ \lambda _ {ij} ^ {a} \} , . . . ; $$

of orders one, two, etc. In the general case there exists a fundamental geometric object

$$ \{ \lambda _ {i} ^ {a} \dots \lambda _ {i _ {1} \dots i _ {k} } ^ {a} \} $$

of finite order $ k $ determining the submanifold $ \mathfrak M _ {m} $ up to constants. In the study of submanifolds of the manifold $ M $, Cartan's method of exterior forms is usually applied in conjunction with the moving-frame method (see, for example, [4]).

The method is named after E. Cartan, who, from 1899 onward, made extensive use of exterior forms.

References

[1]	E. Cartan, "Les systèmes différentiells extérieurs et leurs application en géométrie" , Hermann (1945)
[2]	S.P. Finikov, "Cartan's method of exterior forms in differential geometry" , 1–3 , Moscow-Leningrad (1948) (In Russian)
[3]	S. Helgason, "Differential geometry and symmetric spaces" , Acad. Press (1962) MR0145455 Zbl 0111.18101
[4]	E. Cartan, "La géométrie des éspaces Riemanniennes" , Mém. Sci. Math. , 9 , Gauthier-Villars (1925)

Comments

It is more usual to view an exterior form of degree $ p $ as a function $ \Omega : E ^ {p} \rightarrow K $ such that: a) $ v _ {i} \mapsto \Omega (v _ {1} \dots v _ {i} \dots v _ {p} ) $ is linear (from $ E $ to $ K $) for each $ i $; and b) $ \Omega (v _ {1} \dots v _ {i} \dots v _ {j} \dots v _ {p} ) = 0 $ if $ v _ {i} = v _ {j} $, $ i \neq j $. Here $ E $ is a fixed vector space over a field $ K $ of dimension $ n $.

If $ e _ {1} \dots e _ {n} $ is a basis of $ E $, then $ e ^ {i _ {1} \dots i _ {p} } $, $ i _ {1} < \dots < i _ {p} $, is the exterior form which assigns the value 1 to $ (e _ {i _ {1} } \dots e _ {i _ {p} } ) $ and the value 0 to any $ p $- tuple of basis vectors containing some $ e _ {j} $ with $ j \notin \{ i _ {1} \dots i _ {p} \} $.

Taking for $ E $ the tangent space of a manifold $ M $ at a point $ x \in M $ and $ K = \mathbf R $, one gets the link with the article's description of the Cartan calculus on manifolds.

The inner product, or contraction, of $ \Omega $ with the vector $ v \in E $ is the exterior form of degree $ p - 1 $ given by $ (v _ {1} \dots v _ {p - 1 } ) \mapsto \Omega (v, v _ {1} \dots v _ {p} ) $; it is denoted by $ v \llcorner \Omega $ or $ i (v) \cdot \Omega $. The "first-order algebraic derivative" $ {\partial \Omega _ {p} } / {\partial e ^ {i} } $ of the article is equal to $ e _ {i} \llcorner \Omega _ {p} $.

In the definition of differential Grassmann algebras, the set $ K $ of analytic functions is not a field. However, no problems arise if one uses rings instead, and in fact the ring of $ C ^ \infty $ functions is used in the article when discussing the Cartan calculus on manifolds.

Instead of $ D \Omega _ {p} $ for the exterior differential of a $ p $- form $ \Omega _ {p} $ one more often uses the notation $ d \Omega _ {p} $. The exterior differential of a function $ f $ is $ df = ( \partial f / \partial x ^ {i} ) dx ^ {i} $.

In the Western literature Ostrogradski's theorem is usually called the Stokes theorem; the statement that a Pfaffian form is locally the total derivative of a function if and only if its exterior derivative is zero is of course (part) of the Poincaré lemma; cf. also Differential form for these two items.

For a complete account on systems of Pfaffian equations, including the Cartan–Kähler theorem and the Cartan–Kuranishi theorem, see [a1] and Pfaffian structure; Pfaffian equation and Pfaffian problem.

The Grassmann algebra attached to a vector space $ V $ is the special case of the Clifford algebra attached to a vector space $ V $ and a quadratic form $ Q $ in case $ Q = 0 $.

References

[a1]	J. Dieudonné, "Treatise on analysis" , Acad. Press (1974) pp. Chapt. 18, Sect. 8–14 (Translated from French) MR0362066 Zbl 0292.58001
[a2]	E. Cartan, "Leçons sur les invariants intégraux" , Hermann (1971) MR0355764 Zbl 0212.12501