The geometry of multiple integrals depending on parameters

We consider the geometry of an integral in the situation where the multiple integral depends on a finite number of parameters; the problems of exhibiting properties invariant with respect to certain classes of transformations (involving the underlying manifold and the parameter manifold) and the corresponding classification of integrals are solved by the methods of modern differential geometry.Translated fromItogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 22, 1990, pp. 37–58.