×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] 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).
[2] 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).
[3] S. Kh. Arutyunyan, ?The geometry of ann-fold integral depending onn+1 parameters,? in:Differential Geometry [in Russian], Kalinin (1977), pp. 23?34.
[4] 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).
[5] 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).
[6] S. Kh. Arutyunyan, ?The geometry of an (n +s)-fold integral depending onn parameters,?Izv. Vuzov, Mat., No. 11, 3?10 (1984).
[7] S. Kh. Arutyunyan, ?The geometry of an (n+1)-fold integral depending onn parameters,?Izv. Vuzov, Mat., No. 3, 6?13 (1987).
[8] V. I. Bliznikas, ?Finsler spaces and their generalizations,?Itogi Nauk i Tekhniki. Algebra. Topology. Geometry, 73?125 (1967).
[9] G. Busernann,The Geometry of Geodesies, Academic Press, New York (1955).
[10] 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).
[11] A. M. Vasil’ev, ?Differential algebra as a method of differential geometry,?Tr. Geometr. Sem., Inst. Inf. AN SSSR,1, 33?61 (1966).
[12] A. M. Vasil’ev,The Theory of Differential-Geometric Structures [in Russian], Moscow University Press (1987). · Zbl 0643.70011
[13] V. V. Vishnevskii, ?On a generalization of the Shirokov-Rashevskii spaces,?Uch. Zap. Kazansk. Univ.,125, No. 1, 60?73 (1965).
[14] V. V. Vishnevskii, ?On a parabolic analog of the A-spaces,?Izv. Vuzov, Mat., No. 1, 29?38 (1968).
[15] E. Cartan,Spaces of Affine, Projective, and Conformai Connection [Russian translation], Kazan University Press, (1962).
[16] 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).
[17] I. G. Mulin, ?On spaces of Shirokov type,? in:Global and Riemannian Geometry [in Russian], Leningrad (1983), pp. 62?66.
[18] A. P. Norden, ?On a class of four-dimensional A-spaces,?Izv. Vuzov, Mat., No. 4, 145?157 (1960).
[19] A. P. Norden, ?Cartesian composition spaces,?Izv. Vuzov, Mat., No. 4, 117?128 (1963).
[20] 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).
[21] A. Z. Petrov,Einstein Spaces, Pergamon Press, New York (1969). · Zbl 0174.28305
[22] Yu. G. Petrov, ?Some generalizations of the Shirokov-Rashevskii spaces,?Uch. Zap. Chuvash. Gos. Fed. Inst., No. 29, 50?77 (1969).
[23] Yu. G. Petrov, ?On the realization of complex Weyl spaces,?Izv. Vuzov, Mat., No. 6, 112?118 (1977).
[24] P. K. Rashevskii, ?The scalar field in a stratified space,?Tr. Sem. po Vekt. i Tenz. Anal, No. 6 (1948).
[25] 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).
[26] B. A. Rozenfel’d, ?On unitary and stratified spaces,?Tr. Sem. po Vekt. i Tenz. Anal, No. 7 (1949).
[27] B. A. Rozenfel’d, ?The projective-differential geometry of families of pairsPm+Pn-m-1 inPn, Mat. Sb.,24(66), No. 3 (1949).
[28] S. Sternberg,Lectures on Differential Geometry, Chelsea, New York (1983).
[29] 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).
[30] A. S. Fedenko, ?Limiting Spaces,?Usp. Mat. Nauk,12, No. 3, 235?240 (1957).
[31] A. S. Fedenko,Spaces with Symmetries [in Russian], Belorus. University Press, Minsk (1977). · Zbl 0463.53034
[32] S. P. Finikov,Theorie der Kongruenzen, Akademie-Verlag, Berlin (1959).
[33] P. A. Shirokov, ?Constant fields of vectors and tensors in Riemannian spaces,?Izv. Kazan. Fiz.-Mai. Obshch., Ser. 2,25, 86?114 (1925).
[34] P. A. Shirokov, ?On a type of symmetric space,?Mat. Sb.,41, No. 3, 362?372 (1957).
[35] P. A. Shirokov and A. P. Shirokov,Affine Differential Geometry [in Russian], Fizmatgiz, Moscow (1959). · Zbl 0085.36701
[36] A. Avez, ?Conditions nécessaires et suffisantes pour qu’une variété soit un espace d’Einstein,?C. R.,248, No. 8, 1113?1115 (1959).
[37] A. Besse,Einstein Manifolds, Springer, Berlin (1987).
[38] E. Cartan,Les espaces métriques fondés sur la notion d’aire, Paris (1933).
[39] E. Cartan, ?La géométrie de l’intégrale ?F(r,y,y?, y?)dx? ?uvres Complètes, III, (1941), p. 2.
[40] P. Dedecker, ?On the generalization of symplectic geometry to multiple integrals in the calculus of variations,?Led. Notes Math., No. 570, 395?456 (1977). · Zbl 0352.49018 · doi:10.1007/BFb0087794
[41] L. P. Eisenhart, ?Spaces for which the Ricci scalarR is equal to zero,?Proc. Nat. Acad. Sci. USA,44, No. 7, 695?698 (1958). · Zbl 0081.15802 · doi:10.1073/pnas.44.7.695
[42] L. P. Eisenhart, ?Spaces for which the Ricci scalarR is equal to zero,?Proc. Nat. Acad. Sci. USA,45, No. 2, 226?229 (1959). · Zbl 0084.18302 · doi:10.1073/pnas.45.2.226
[43] J. Géhéniau, ?Une classification des espaces einsteiniennes,?C. R.,244, No. 6, 723?724 (1957).
[44] A. Kawaguchi, ?Theory of connections in the generalized Finsler manifold,?Proc. Imp. Acad. Tokyo,7, 211?214 (1937). · Zbl 0002.15801 · doi:10.3792/pia/1195581142
[45] 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). · Zbl 0019.27801
[46] E. Kahler, ?Über eine bemerkenswerte Hermitesche Metrik,?Abh. Math. Sem. Hamburg Univ., B. 9 (1933).
[47] G. Ludwig and G. Scanlan, ?Classification of the Ricci tensor,?Comm. Math. Phys.,20, No. 4, 291?300 (1971). · Zbl 0207.20702 · doi:10.1007/BF01646625
[48] A. Z. Petrov,Perspectives in Geometry and Relativity, Indiana University Press (1966).
[49] 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). · Zbl 0283.53046
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.