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**The method of equivalence and its applications.**
*(English)*
Zbl 0694.53027

CBMS-NSF Regional Conference Series in Applied Mathematics, 58. Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics. vii, 127 p. $ 21.75 (1989).

The booklet provides a thorough, clear, and carefully written introduction into E. Cartan’s method of equivalence presented for the first time from the point of view both of geometry and of applied mathematical analysis. The stress is laid on the algorithmical description conveniently illustrated by a number of examples selected from the calculus of variations, control theory, ordinary and partial differential equations, and Riemannian geometry. Only a modest background in calculus on manifolds and elements of Lie groups are assumed.

In general, the equivalence problem consists in determining the conditions under which two given objects of the same nature can be mapped onto each other. After Cartan, it can be reduced to the following standard form: given on the one hand a coframe \(\omega_ 1,...,\omega_ n\) (a basis of the cotangent space) in the variables \(x_ 1,...,x_ n\), on the other hand a coframe \(\vartheta_ 1,...,\vartheta_ n\) in the variables \(y_ 1,...,y_ n,\) determine if there exists a diffeomorphism \(f: x_ i\equiv f_ i(y_ 1,...,y_ n)\quad (i=1,...,m),\) \(x_ i\equiv y_ i\) \((i=m+1,...,n)\) such that \(f^*\vartheta_ j\equiv \sum g^ k_ j\omega_ k\) where \(g=(g^ k_ j)\) belongs to a prescribed linear group G depending possibly on the parameters \(x_{m+1},...,x_ n\) (here m is given, \(m\leq n)\). The author deals only with the particular case of constant G (i.e., \(m=n)\) which permits to employ the achievements of the popular theory of G-spaces. Unfortunately, following this (common) approach, the results became less general than in the original Cartan’s papers and the calculations may lead into an impass (cf. the author’s remark on p. 37).

Concerning the contents of the book, the introduction consists of interesting historical comments, chapters 1 and 2 specify the equivalence problem and its reformulation in terms of the G-spaces, chapters 3, 4, and 5 are devoted to the technique of reduction of the structure tensor, chapter 6 concerns the elementary subcase \(G=\{e\}\) related to the Lie groups and the Frobenius theorem and, on the contrary, chapters 7 and 10 outline the difficulties in the general case (prolongations, involutiveness, Cartan-Kähler theory) but without any mention of pseudogroups (!), the intermediate chapters 8 and 9 discuss the equivalence for the control systems \(dx/dt=f(x,u)\) under the mappings \(t\to t\), \(x\to \phi (x)\), \(u\to \psi (x,u)\). The appendix presents a short excerpt from E. Cartan’s work concerning the most general equivalence problem and its complete solution.

In general, the equivalence problem consists in determining the conditions under which two given objects of the same nature can be mapped onto each other. After Cartan, it can be reduced to the following standard form: given on the one hand a coframe \(\omega_ 1,...,\omega_ n\) (a basis of the cotangent space) in the variables \(x_ 1,...,x_ n\), on the other hand a coframe \(\vartheta_ 1,...,\vartheta_ n\) in the variables \(y_ 1,...,y_ n,\) determine if there exists a diffeomorphism \(f: x_ i\equiv f_ i(y_ 1,...,y_ n)\quad (i=1,...,m),\) \(x_ i\equiv y_ i\) \((i=m+1,...,n)\) such that \(f^*\vartheta_ j\equiv \sum g^ k_ j\omega_ k\) where \(g=(g^ k_ j)\) belongs to a prescribed linear group G depending possibly on the parameters \(x_{m+1},...,x_ n\) (here m is given, \(m\leq n)\). The author deals only with the particular case of constant G (i.e., \(m=n)\) which permits to employ the achievements of the popular theory of G-spaces. Unfortunately, following this (common) approach, the results became less general than in the original Cartan’s papers and the calculations may lead into an impass (cf. the author’s remark on p. 37).

Concerning the contents of the book, the introduction consists of interesting historical comments, chapters 1 and 2 specify the equivalence problem and its reformulation in terms of the G-spaces, chapters 3, 4, and 5 are devoted to the technique of reduction of the structure tensor, chapter 6 concerns the elementary subcase \(G=\{e\}\) related to the Lie groups and the Frobenius theorem and, on the contrary, chapters 7 and 10 outline the difficulties in the general case (prolongations, involutiveness, Cartan-Kähler theory) but without any mention of pseudogroups (!), the intermediate chapters 8 and 9 discuss the equivalence for the control systems \(dx/dt=f(x,u)\) under the mappings \(t\to t\), \(x\to \phi (x)\), \(u\to \psi (x,u)\). The appendix presents a short excerpt from E. Cartan’s work concerning the most general equivalence problem and its complete solution.

Reviewer: J.Chrastina

### MSC:

53C10 | \(G\)-structures |

58H05 | Pseudogroups and differentiable groupoids |

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |