Cartan's equivalence method
Encyclopedia
In mathematics
, Cartan's equivalence method is a technique in differential geometry for determining whether two geometrical structures are the same up to a diffeomorphism
. For example, if M and N are two Riemannian manifold
s with metrics g and h, respectively,
when is there a diffeomorphism
such that
?
Although the answer to this particular question was known in dimension 2 to Gauss
and in higher dimensions to Christoffel and perhaps Riemann as well, Élie Cartan
and his intellectual heirs developed a technique for answering similar questions for radically different geometric structures. (For example see the Cartan-Karlhede algorithm
.)
Cartan successfully applied his equivalence method to many such structures, including projective structures, CR structures, and complex structures, as well as ostensibly non-geometrical structures such as the equivalence of Lagrangian
s and ordinary differential equation
s. (His techniques were later developed more fully by many others, such as D. C. Spencer and Shiing-Shen Chern
.)
The equivalence method is an essentially algorithm
ic procedure for determining when two geometric structures are identical. For Cartan, the primary geometrical information was expressed in a coframe
or collection of coframes on a differentiable manifold
. See method of moving frames.
for a structure group G. This amounts to giving a special class of coframes on M and N. Cartan's method addresses the question of whether there exists a local diffeomorphism φ:M→N under which the G-structure on N pulls back to the given G-structure on M. An equivalence problem has been "solved" if one can give a complete set of structural invariants for the G-structure: meaning that such a diffeomorphism exists if and only if all of the structural invariants agree in a suitably defined sense.
Explicitly, local systems of one-forms θi and γi are given on M and N, respectively, which span the respective cotangent bundles (i.e., are coframe
s). The question is whether there is a local diffeomorphism φ:M→N such that the pullback of the coframe on N satisfies
(1)
where the coefficient g is a function on M taking values in the Lie group
G. For example, if M and N are Riemannian manifolds, then G=O(n) is the orthogonal group and θi and γi are orthonormal coframes of M and N respectively. The question of whether or not two Riemannian manifolds are isometric is then a question of whether there exists a diffeomorphism φ satisfying (1).
The first step in the Cartan method is to express the pullback relation (1) in as invariant a way as possible through the use of a "prolongation". The most economical way to do this is to use a G-subbundle PM of the principal bundle of linear coframes LM, although this approach can lead to unnecessary complications when performing actual calculations. In particular, later on this article uses a different approach. But for the purposes of an overview, it is convenient to stick with the principal bundle viewpoint.
The second step is to use the diffeomorphism invariance of the exterior derivative
to try to isolate any other higher-order invariants of the G-structure. Basically one obtains a connection in the principal bundle PM, with some torsion. The components of the connection and of the torsion are regarded as invariants of the problem.
The third step is that if the remaining torsion coefficients are not constant in the fibres of the principal bundle PM, it is often possible (although sometimes difficult), to normalize them by setting them equal to a convenient constant value and solving these normalization equations, thereby reducing the effective dimension of the Lie group G. If this occurs, one goes back to step one, now having a Lie group of one lower dimension to work with.
Complete reduction. Here the structure group has been reduced completely to the trivial group
. The problem can now be handled by methods such as the Frobenius theorem
. In other words, the algorithm has successfully terminated.
On the other hand, it is possible that the torsion coefficients are constant on the fibres of PM. Equivalently, they no longer depend on the Lie group G because there is nothing left to normalize, although there may still be some torsion. The three remaining cases assume this.
Involution. The equivalence problem is said to be involutive (or in involution) if it passes Cartan's test. This is essentially a rank condition on the connection obtained in the first three steps of the procedure. The Cartan test generalizes the Frobenius theorem
on the solubility of first-order linear systems of partial differential equations. If the coframes on M and N (obtained by a thorough application of the first three steps of the algorithm) agree and satisfy the Cartan test, then the two G-structures are equivalent. (Actually, to the best of the author's knowledge, the coframes must be real analytic in order for this to hold, because the Cartan-Kähler theorem requires analyticity.)
Prolongation. This is the most intricate case. In fact there are two sub-cases. In the first sub-case, all of the torsion can be uniquely absorbed into the connection form. (Riemannian manifolds are an example, since the Levi-Civita connection absorbs all of the torsion). The connection coefficients and their invariant derivatives form a complete set of invariants of the structure, and the equivalence problem is solved. In the second subcase, however, it is either impossible to absorb all of the torsion, or there is some ambiguity (as is often the case in Gaussian elimination
, for example). Here, just as in Gaussian elimination, there are additional parameters which appear in attempting to absorb the torsion. These parameters themselves turn out to be additional invariants of the problem, so the structure group G must be prolonged into a subgroup of a jet group
. Once this is done, one obtains a new coframe on the prolonged space and has to return to the first step of the equivalence method. (See also prolongation of G-structures.)
Degeneracy. Because of a non-uniformity of some rank condition, the equivalence method is unsuccessful in handling this particular equivalence problem. For example, consider the equivalence problem of mapping a manifold M with a single one-form θ to another manifold with a single one-form γ such that φ*γ=θ. The zeros of these one forms, as well as the rank of their exterior derivatives at each point need to be taken into account. The equivalence method can handle such problems if all of the ranks are uniform, but it is not always suitable if the rank changes. Of course, depending on the particular application, a great deal of information can still be obtained with the equivalence method.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Cartan's equivalence method is a technique in differential geometry for determining whether two geometrical structures are the same up to a diffeomorphism
Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...
. For example, if M and N are two Riemannian manifold
Riemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
s with metrics g and h, respectively,
when is there a diffeomorphism
such that
?Although the answer to this particular question was known in dimension 2 to Gauss
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
and in higher dimensions to Christoffel and perhaps Riemann as well, Élie Cartan
Élie Cartan
Élie Joseph Cartan was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications...
and his intellectual heirs developed a technique for answering similar questions for radically different geometric structures. (For example see the Cartan-Karlhede algorithm
Cartan-Karlhede algorithm
One of the most fundamental problems of Riemannian geometry is this: given two Riemannian manifolds of the same dimension, how can one tell if they are locally isometric? This question was addressed by Elwin Christoffel, and completely solved by Élie Cartan using his exterior calculus with his...
.)
Cartan successfully applied his equivalence method to many such structures, including projective structures, CR structures, and complex structures, as well as ostensibly non-geometrical structures such as the equivalence of Lagrangian
Lagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...
s and ordinary differential equation
Ordinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
s. (His techniques were later developed more fully by many others, such as D. C. Spencer and Shiing-Shen Chern
Shiing-Shen Chern
Shiing-Shen Chern was a Chinese American mathematician, one of the leaders in differential geometry of the twentieth century.-Early years in China:...
.)
The equivalence method is an essentially algorithm
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...
ic procedure for determining when two geometric structures are identical. For Cartan, the primary geometrical information was expressed in a coframe
Coframe
In mathematics, a coframe or coframe field on a smooth manifold M is a system of one-forms or covectors which form a basis of the cotangent bundle at every point. In the exterior algebra of M, one has a natural map from v_k:\bigoplus^kT^*M\to\bigwedge^kT^*M, given by v_k:\mapsto...
or collection of coframes on a differentiable manifold
Differentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
. See method of moving frames.
Overview of Cartan's method
Specifically, suppose that M and N are a pair of manifolds each carrying a G-structureG-structure
In differential geometry, a G-structure on an n-manifold M, for a given structure group G, is a G-subbundle of the tangent frame bundle FM of M....
for a structure group G. This amounts to giving a special class of coframes on M and N. Cartan's method addresses the question of whether there exists a local diffeomorphism φ:M→N under which the G-structure on N pulls back to the given G-structure on M. An equivalence problem has been "solved" if one can give a complete set of structural invariants for the G-structure: meaning that such a diffeomorphism exists if and only if all of the structural invariants agree in a suitably defined sense.
Explicitly, local systems of one-forms θi and γi are given on M and N, respectively, which span the respective cotangent bundles (i.e., are coframe
Coframe
In mathematics, a coframe or coframe field on a smooth manifold M is a system of one-forms or covectors which form a basis of the cotangent bundle at every point. In the exterior algebra of M, one has a natural map from v_k:\bigoplus^kT^*M\to\bigwedge^kT^*M, given by v_k:\mapsto...
s). The question is whether there is a local diffeomorphism φ:M→N such that the pullback of the coframe on N satisfies
(1)where the coefficient g is a function on M taking values in the Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
G. For example, if M and N are Riemannian manifolds, then G=O(n) is the orthogonal group and θi and γi are orthonormal coframes of M and N respectively. The question of whether or not two Riemannian manifolds are isometric is then a question of whether there exists a diffeomorphism φ satisfying (1).
The first step in the Cartan method is to express the pullback relation (1) in as invariant a way as possible through the use of a "prolongation". The most economical way to do this is to use a G-subbundle PM of the principal bundle of linear coframes LM, although this approach can lead to unnecessary complications when performing actual calculations. In particular, later on this article uses a different approach. But for the purposes of an overview, it is convenient to stick with the principal bundle viewpoint.
The second step is to use the diffeomorphism invariance of the exterior derivative
Exterior derivative
In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....
to try to isolate any other higher-order invariants of the G-structure. Basically one obtains a connection in the principal bundle PM, with some torsion. The components of the connection and of the torsion are regarded as invariants of the problem.
The third step is that if the remaining torsion coefficients are not constant in the fibres of the principal bundle PM, it is often possible (although sometimes difficult), to normalize them by setting them equal to a convenient constant value and solving these normalization equations, thereby reducing the effective dimension of the Lie group G. If this occurs, one goes back to step one, now having a Lie group of one lower dimension to work with.
The fourth step
The main purpose of the first three steps was to reduce the structure group itself as much as possible. Suppose that the equivalence problem has been through the loop enough times that no further reduction is possible. At this point, there are various possible directions in which the equivalence method leads. For most equivalence problems, there are only four cases: complete reduction, involution, prolongation, and degeneracy.Complete reduction. Here the structure group has been reduced completely to the trivial group
Trivial group
In mathematics, a trivial group is a group consisting of a single element. All such groups are isomorphic so one often speaks of the trivial group. The single element of the trivial group is the identity element so it usually denoted as such, 0, 1 or e depending on the context...
. The problem can now be handled by methods such as the Frobenius theorem
Frobenius theorem (differential topology)
In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system of first-order homogeneous linear partial differential equations...
. In other words, the algorithm has successfully terminated.
On the other hand, it is possible that the torsion coefficients are constant on the fibres of PM. Equivalently, they no longer depend on the Lie group G because there is nothing left to normalize, although there may still be some torsion. The three remaining cases assume this.
Involution. The equivalence problem is said to be involutive (or in involution) if it passes Cartan's test. This is essentially a rank condition on the connection obtained in the first three steps of the procedure. The Cartan test generalizes the Frobenius theorem
Frobenius theorem (differential topology)
In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system of first-order homogeneous linear partial differential equations...
on the solubility of first-order linear systems of partial differential equations. If the coframes on M and N (obtained by a thorough application of the first three steps of the algorithm) agree and satisfy the Cartan test, then the two G-structures are equivalent. (Actually, to the best of the author's knowledge, the coframes must be real analytic in order for this to hold, because the Cartan-Kähler theorem requires analyticity.)
Prolongation. This is the most intricate case. In fact there are two sub-cases. In the first sub-case, all of the torsion can be uniquely absorbed into the connection form. (Riemannian manifolds are an example, since the Levi-Civita connection absorbs all of the torsion). The connection coefficients and their invariant derivatives form a complete set of invariants of the structure, and the equivalence problem is solved. In the second subcase, however, it is either impossible to absorb all of the torsion, or there is some ambiguity (as is often the case in Gaussian elimination
Gaussian elimination
In linear algebra, Gaussian elimination is an algorithm for solving systems of linear equations. It can also be used to find the rank of a matrix, to calculate the determinant of a matrix, and to calculate the inverse of an invertible square matrix...
, for example). Here, just as in Gaussian elimination, there are additional parameters which appear in attempting to absorb the torsion. These parameters themselves turn out to be additional invariants of the problem, so the structure group G must be prolonged into a subgroup of a jet group
Jet group
In mathematics, a jet group is a generalization of the general linear group which applies to Taylor polynomials instead of vectors at a point. Essentially a jet group describes how a Taylor polynomial transforms under changes of coordinate systems .The k-th order jet group Gnk consists of jets of...
. Once this is done, one obtains a new coframe on the prolonged space and has to return to the first step of the equivalence method. (See also prolongation of G-structures.)
Degeneracy. Because of a non-uniformity of some rank condition, the equivalence method is unsuccessful in handling this particular equivalence problem. For example, consider the equivalence problem of mapping a manifold M with a single one-form θ to another manifold with a single one-form γ such that φ*γ=θ. The zeros of these one forms, as well as the rank of their exterior derivatives at each point need to be taken into account. The equivalence method can handle such problems if all of the ranks are uniform, but it is not always suitable if the rank changes. Of course, depending on the particular application, a great deal of information can still be obtained with the equivalence method.