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 be a
-dimensional vector space over an arbitrary field
with basis vectors
,
. In addition to the basis vectors, one defines for any natural number
the vectors
,
, according to the following rule: If at least two of the natural numbers
are identical, then
; if all the
are distinct and the numbers
are a permutation of
, then
if the permutation
,
, is even, and
if this permutation is odd. In the vector space
the exterior product is defined:
; in addition, the usual laws for a hypercomplex system (i.e. an associative algebra) are required to hold. The algebra of dimension
over
so constructed is called the Grassmann algebra. A vector of the form
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is called a monomial of degree ,
. A sum of monomials of the same degree
is called an exterior form of degree
; a sum of monomials of the first degree is called a linear form. The elements of the field
are, by definition, forms of degree zero. The vectors
generate the Grassmann algebra and so do any
linearly independent combinations of them
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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
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of degree with respect to the symbol
is meant the form
of degree
, obtained from
by replacing by zero all monomials not containing the symbol
, while for each of the remaining monomials the symbol
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
-th order non-zero algebraic derivatives of the form
is called the associated system of linear forms of
. The rank of the exterior form
is the rank of its associated system. It is equal to the minimum number of linear forms in terms of which
can be expressed using the exterior product operation. For the study of a system of differential equations in
, the differential Grassmann algebra is used, where
is taken to be the ring of analytic functions in
real variables
defined in some domain of
, and the vectors
are denoted by
. Its linear forms are called
-forms or Pfaffian forms, where the symbols
are the differentials of the variables
. The exterior forms of degree
are called
-forms or exterior differential forms of degree
. By the exterior differential of the
-form
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is meant the -form
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The exterior differential has the following properties:
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where ,
are arbitrary
-forms and
is an arbitrary
-form.
A Pfaffian form is locally the total differential of some function
if and only if its exterior differential vanishes. Let
![]() | (1) |
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be an arbitrary system of linearly independent Pfaffian equations in independent variables
and
unknown functions
. The system
is called the closure of the system (1). The closure is called pure closure (denoted by
) if the original system (1) is algebraically accounted for in it, that is, if the quantities
in (1) are substituted into the quadratic forms
. The system
,
, or the system
,
equivalent to it, is called a closed system. The system (1) is completely integrable if and only if
. Equating to zero the algebraic derivatives of
with respect to
and
,
;
, 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
be the result of substituting for
in the algebraic derivative
the arbitrary variables
,
. Associated with the system (1) is the sequence of matrices
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The numbers
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are called the characteristics and the number
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is called the Cartan number of the system (1). By adjoining to the closed system ,
the equations
, where the
are new unknown functions, one obtains the first prolongation of the system (1). Let
be the number of functionally independent functions among the
. Then always
. If
, then the system (1) is in involution and its general solution depends on
arbitrary functions in
arguments,
functions in
arguments, etc.,
functions in one argument and
arbitrary constants. If, on the other hand,
, 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
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with independent variables and unknown functions
(
,
). Its pure closure has the form:
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For this system:
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The system is not in involution. The prolongated system
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is completely integrable and its general solution has the form
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where 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
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where is an analytic oriented
-dimensional manifold,
is its
-dimensional smooth boundary,
is an
-form, and
is its exterior differential. The formula for the change of variables in a multiple integral
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under a mapping , defined by the formulas
, where
, is obtained by the direct change of the variables
and their differentials
. Since
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it follows that
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Cartan's method of exterior forms is extensively used in the study of manifolds with various structures. Let be a differentiable manifold of class
, let
be the set of differentiable functions on
, let
be the set of all the vector fields on
, and let
be the set of skew-symmetric
-multilinear mappings on the module
(
copies, where
is a natural number).
Let and denote by
the direct sum of the
:
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The elements of the module are called exterior differential forms on
; the elements of
are called
-forms. Let
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Then their exterior product is defined by the formulas:
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where is the group of permutations of the set
, and
or
depending on whether the permutation
is even or odd. The module
of skew-symmetric
-multilinear functions with the exterior product is called the Grassmann algebra over the manifold
. If
is
, then one obtains the differential Grassmann algebra considered earlier. By exterior differentiation one means the
-linear mapping
with the following properties:
for every
; if
, then
is the
-form defined by the formula
, where
;
,
, if
,
. Suppose, for instance, that
is a manifold with a given affine connection. An affine connection on a manifold
is a rule
which associates with each
a linear mapping
of the vector space
into itself, satisfying the following two properties:
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for ,
. The operator
is called the covariant derivative with respect to
. Let
be a diffeomorphism of
, and
an affine connection on
. The formula
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where , defines a new affine connection on
. One says that
is invariant with respect to
if
. In this case
is called an affine transformation of
. Let
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for all , and let
be the module dual to the
-module
. The
-multilinear mapping
, where
is a Pfaffian form, is called the torsion tensor field, and is denoted by
; the
-multilinear mapping
is called the curvature tensor field, and is denoted by
. Let
and let
be a basis for the vector fields in some neighbourhood
of the point
. The functions
,
,
are defined on
by the formulas
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For the -forms
,
defined on
by the formulas
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the following structural equations of Cartan hold:
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The system of Pfaffian equations
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defines an -dimensional submanifold
. Extending this system by means of the Cartan structural equations, one obtains a sequence of fundamental geometric objects of the submanifold
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of orders one, two, etc. In the general case there exists a fundamental geometric object
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of finite order determining the submanifold
up to constants. In the study of submanifolds of the manifold
, 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) |
[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 as a function
such that: a)
is linear (from
to
) for each
; and b)
if
,
. Here
is a fixed vector space over a field
of dimension
.
If is a basis of
, then
,
, is the exterior form which assigns the value 1 to
and the value 0 to any
-tuple of basis vectors containing some
with
.
Taking for the tangent space of a manifold
at a point
and
, one gets the link with the article's description of the Cartan calculus on manifolds.
The inner product, or contraction, of with the vector
is the exterior form of degree
given by
; it is denoted by
or
. The "first-order algebraic derivative"
of the article is equal to
.
In the definition of differential Grassmann algebras, the set of analytic functions is not a field. However, no problems arise if one uses rings instead, and in fact the ring of
functions is used in the article when discussing the Cartan calculus on manifolds.
Instead of for the exterior differential of a
-form
one more often uses the notation
. The exterior differential of a function
is
.
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 is the special case of the Clifford algebra attached to a vector space
and a quadratic form
in case
.
References
[a1] | J. Dieudonné, "Treatise on analysis" , Acad. Press (1974) pp. Chapt. 18, Sect. 8–14 (Translated from French) |
[a2] | E. Cartan, "Leçons sur les invariants intégraux" , Hermann (1971) |
Cartan method of exterior forms. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cartan_method_of_exterior_forms&oldid=19174