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The geometry of multiple integrals depending on parameters. (English. Russian original) Zbl 0737.53030
J. Sov. Math. 55, No. 5, 1954-1969 (1991); translation from Itogi Nauki Tekh., Ser. Probl. Geom. 22, 37-58 (1990).
Summary: The problem of constructing a geometry for integration was first posed by E. Cartan [Les espaces métriques fondés sur la notion d’aire (Paris 1933; Zbl 0008.27202)] and in its general form consisted of studying the differential-geometric structures that can be attached invariantly to an integral on the underlying manifold. An important role in this field has been played by the works [E. Kähler, Abh. Math. Semin. Univ. Hamb. 9, 173-186 (1932; Zbl 0005.41301)], [G. Ludwig and G. Scanlan, Commun. Math. Phys. 20, 291-300 (1971; Zbl 0207.207)], of Kawaguchi, who introduced the spaces that later came to bear his name. Despite the very large number of papers, many aspects of the construction of a geometric theory for multiple integrals are not yet completely worked out (cf., for example, [P. Dedecker, On the generalization of symplectic geometry to multiple integrals in the calculus of variations, Lect. Notes Math. 570, 395-456 (1977; Zbl 0352.49018)]). A survey of these investigations can be found in the paper of V. I. Bliznikas [Itogi Nauki, Ser. Mat., Algebra, Topologiya, Geom. 73-125 (1969; Zbl 0228.53039)].
An analogous formulation of the problem can be maintained also in the case of multiple integrals depending on parameters – the study of differential-geometric structures defined on the manifold of the variables of integration and the parameters for these integrals. In the systematic study of such integrals, as well as the integral transforms defined by them, it becomes necessary to exhibit properties of the integrals that are independent of the choice of coordinates on the manifold over which the integration is carried out and the choice of coordinates on the manifold of parameters. Further, in the theory of integral transforms there developed some interest in the properties of integrals that are preserved when the kernel is multiplied by a function of the variables of integration alone or of the parameters alone: $K(x,y)\to f(x)g(y)K(x,y).$ The problem of exhibiting such invariant properties and the corresponding classification of integrals is typical for the modern theory of differential-geometric structures and can be solved by the methods of modern differential-geometric investigation whose origins date back to the work of E. Cartan [J. Math. Pures Appl., IX, Ser. 15, 42-69 (1936; Zbl 0014.36801)] and were propagated in the USSR and significantly developed by S. P. Finikov [Theorie der Kongruenzen (Berlin, 1959; Zbl 0085.367)], G. F. Laptev [Tr. Mosk. Mat. 0.-va 2, 275-382 (1953; Zbl 0053.428)], and A. M. Vasil’ev [Tr. Geom. Semin. 1, 33-61 (1966; Zbl 0163.441)], and [Theory of differential-geometric structures (Moskow 1987; Zbl 0656.53001)].
##### MSC:
 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 53C65 Integral geometry 53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
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