Abstract
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.
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Literature cited
S. Kh. Arutyunyan, “The geometry of an n-fold integral depending onn parameters,”Aikakan SSR Gitutyunneri Akademia, Zekuitsner, Dokl. AN ArmSSR,61, No. 1, 7–14 (1975).
S. Kh. Arutyunyan, “The geometry of n-fold integrals depending onn + s parameters,”Aikakan, SSR Gitutyunneri Akademia, Zekuitsner, Dokl. AN ArmSSR,62, No. 1, 15–22 (1976).
S. Kh. Arutyunyan, “The geometry of ann-fold integral depending onn+1 parameters,” in:Differential Geometry [in Russian], Kalinin (1977), pp. 23–34.
S. Kh. Arutyunyan, “On the geometry of the symmetric space of pairs of points ofn-dimensional conformai space,”Aikakan SSR Gitutyunneri Akademia, Zekuitsner, Dokl. AN ArmSSR,71, No. 2, 69–75 (1980).
S. Kh. Arutyunyan, “On the geometry of the symmetric space of null-pairs of the projective spaceRP n,”Aikakan SSR Gitutyunneri Akademia, Dokl. AN ArmSSR,72, No. 4, 203–210 (1981).
S. Kh. Arutyunyan, “The geometry of an (n +s)-fold integral depending onn parameters,”Izv. Vuzov, Mat., No. 11, 3–10 (1984).
S. Kh. Arutyunyan, “The geometry of an (n+1)-fold integral depending onn parameters,”Izv. Vuzov, Mat., No. 3, 6–13 (1987).
V. I. Bliznikas, “Finsler spaces and their generalizations,”Itogi Nauk i Tekhniki. Algebra. Topology. Geometry, 73–125 (1967).
G. Busernann,The Geometry of Geodesies, Academic Press, New York (1955).
A. M. Vasil'ev, “Systems of three first-order partial differential equations in three unknown functions and two independent variables (the local theory),”Mat. Sb.,70, No. 4, 457–480 (1966).
A. M. Vasil'ev, “Differential algebra as a method of differential geometry,”Tr. Geometr. Sem., Inst. Inf. AN SSSR,1, 33–61 (1966).
A. M. Vasil'ev,The Theory of Differential-Geometric Structures [in Russian], Moscow University Press (1987).
V. V. Vishnevskii, “On a generalization of the Shirokov-Rashevskii spaces,”Uch. Zap. Kazansk. Univ.,125, No. 1, 60–73 (1965).
V. V. Vishnevskii, “On a parabolic analog of the A-spaces,”Izv. Vuzov, Mat., No. 1, 29–38 (1968).
E. Cartan,Spaces of Affine, Projective, and Conformai Connection [Russian translation], Kazan University Press, (1962).
G. F. Laptev, “The differential geometry of immersed manifolds. A group-theoretic method for geometric investigation,”Tr. Mosk. Mat. Obshch., No. 2, 275–382 (1953).
I. G. Mulin, “On spaces of Shirokov type,” in:Global and Riemannian Geometry [in Russian], Leningrad (1983), pp. 62–66.
A. P. Norden, “On a class of four-dimensional A-spaces,”Izv. Vuzov, Mat., No. 4, 145–157 (1960).
A. P. Norden, “Cartesian composition spaces,”Izv. Vuzov, Mat., No. 4, 117–128 (1963).
A. P. Norden, “On the structure of the connection on the manifold of lines of a non-Euclidean space,”Izv. Vuzov, Mat., No. 12, 84–94 (1972).
A. Z. Petrov,Einstein Spaces, Pergamon Press, New York (1969).
Yu. G. Petrov, “Some generalizations of the Shirokov-Rashevskii spaces,”Uch. Zap. Chuvash. Gos. Fed. Inst., No. 29, 50–77 (1969).
Yu. G. Petrov, “On the realization of complex Weyl spaces,”Izv. Vuzov, Mat., No. 6, 112–118 (1977).
P. K. Rashevskii, “The scalar field in a stratified space,”Tr. Sem. po Vekt. i Tenz. Anal, No. 6 (1948).
P. K. Rashevskii, “On a pair of connections onn-dimensional surfaces in a 2n-dimensional stratified space,”Tr. Sem. po Vekt. i Tenz. Anal, No. 8 (1950).
B. A. Rozenfel'd, “On unitary and stratified spaces,”Tr. Sem. po Vekt. i Tenz. Anal, No. 7 (1949).
B. A. Rozenfel'd, “The projective-differential geometry of families of pairsPm+Pn-m-1 inPn, Mat. Sb.,24(66), No. 3 (1949).
S. Sternberg,Lectures on Differential Geometry, Chelsea, New York (1983).
L. Tuulmets, “On the geometry of a homogeneous space ofm-pairs and its manifold,”Tastu Ülikooli toimetised, Uch. Zap. Tartus. Univ. No. 464/22, 98–115 (1978).
A. S. Fedenko, “Limiting Spaces,”Usp. Mat. Nauk,12, No. 3, 235–240 (1957).
A. S. Fedenko,Spaces with Symmetries [in Russian], Belorus. University Press, Minsk (1977).
S. P. Finikov,Theorie der Kongruenzen, Akademie-Verlag, Berlin (1959).
P. A. Shirokov, “Constant fields of vectors and tensors in Riemannian spaces,”Izv. Kazan. Fiz.-Mai. Obshch., Ser. 2,25, 86–114 (1925).
P. A. Shirokov, “On a type of symmetric space,”Mat. Sb.,41, No. 3, 362–372 (1957).
P. A. Shirokov and A. P. Shirokov,Affine Differential Geometry [in Russian], Fizmatgiz, Moscow (1959).
A. Avez, “Conditions nécessaires et suffisantes pour qu'une variété soit un espace d'Einstein,”C. R.,248, No. 8, 1113–1115 (1959).
A. Besse,Einstein Manifolds, Springer, Berlin (1987).
E. Cartan,Les espaces métriques fondés sur la notion d'aire, Paris (1933).
E. Cartan, “La géométrie de l'intégrale ∫F(r,y,y′, y″)dx” Œuvres Complètes, III, (1941), p. 2.
P. Dedecker, “On the generalization of symplectic geometry to multiple integrals in the calculus of variations,”Led. Notes Math., No. 570, 395–456 (1977).
L. P. Eisenhart, “Spaces for which the Ricci scalarR is equal to zero,”Proc. Nat. Acad. Sci. USA,44, No. 7, 695–698 (1958).
L. P. Eisenhart, “Spaces for which the Ricci scalarR is equal to zero,”Proc. Nat. Acad. Sci. USA,45, No. 2, 226–229 (1959).
J. Géhéniau, “Une classification des espaces einsteiniennes,”C. R.,244, No. 6, 723–724 (1957).
A. Kawaguchi, “Theory of connections in the generalized Finsler manifold,”Proc. Imp. Acad. Tokyo,7, 211–214 (1937).
A. Kawaguchi, “Geometry in ann-dimensional space with lengths=∫(Ai(x,x′)x″i + B(x,x′1/p dt,”Trans. Amer. Math. Soc.,44, 153–167 (1938).
E. Kahler, “Über eine bemerkenswerte Hermitesche Metrik,”Abh. Math. Sem. Hamburg Univ., B. 9 (1933).
G. Ludwig and G. Scanlan, “Classification of the Ricci tensor,”Comm. Math. Phys.,20, No. 4, 291–300 (1971).
A. Z. Petrov,Perspectives in Geometry and Relativity, Indiana University Press (1966).
R. Rosca, “Variétés pseudo-riemanniennesV n,n de signatre (n, n) et à connexion self-orthogonale involutive,”C. R.,277, No. 19, A959-A961 (1973).
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Translated fromItogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 22, 1990, pp. 37–58.
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Arutyunyan, S.k. The geometry of multiple integrals depending on parameters. J Math Sci 55, 1954–1969 (1991). https://doi.org/10.1007/BF01095671
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DOI: https://doi.org/10.1007/BF01095671