Abstract
The structure of recurrently related operators of the curvature tensor and of its covariant derivatives for n-dimensional Riemannian spaces with arbitrary signature is examined. Applications to Einstein's theory of gravitation are given.
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Literature cited
V. N. Abdullin, “Symmetric Riemannian spaces V4” Izv. Vyssh. Uchebn. Zaved., Mat., No. 2, 3–12 (1971).
D. V. Alekseevskii, “Holonomy groups and recurrent tensor fields in Lorentzian spaces,” in: Problems in Theory of Gravitation and Elementary Particles [in Russian], Vol. 5, Atomizdat, Moscow (1974), pp. 5–17.
V. V. Astrakhantsev, “Pseudo-Riemannian symmetric spaces with commutative holonomy group,” Mat. Sb.,90, No. 2, 288–305 (1973).
S. P. Gavrilov, “Riemannian spaces with one-recurrent bivector. Part II,” in: Gravitation and Relativity Theory [in Russian], Vols. 10–11, Kazan Univ. (1975–76), pp. 158–171.
L. B. Grigor'eva and V. D. Zakharov, “Spaces of recurrent curvature in GRT,” Dokl. Akad. Nauk SSSR,207, No. 4 (1972).
V. D. Zakharov, Gravitational Waves in Einstein's Theory of Gravitation [in Russian], Nauka, Moscow (1972).
V. R. Kaigorodov, “Nonconformally flat relativistic spaces of recurrent curvature,” in: Gravitation and Relativity Theory [in Russian], Vol. 8, Kazan Univ. (1971), pp. 109–115.
V. R. Kaigorodov, “On two-symmetric and two-recurrent gravitational fields,” Preprint Inst. Teor. Fiz. Akad. Nauk SSSR, 72–46R (1972).
V. R. Kaigorodov, “On the curvature of s-recurrent and quasisymmetric Riemannian manifolds,” Dokl. Akad. Nauk SSSR,212, No. 4, 796–799 (1973).
V. R. Kaigorodov, “Riemannian spacesD n Structure of curvature of spaces of type A,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 5, 117–127 (1974).
V. R. Kaigorodov, “Structure of curvature ofD n -spaces of type B,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 1, 104–107 (1975).
V. R. Kaigorodov, “Riemannian spacesD n . Recurrence of second order,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 2, 112–115 (1975).
V. R. Kaigorodov, “Recurrent manifolds and their applications in the theory of holonomy groups,” in: All-Union Scientific Conference on Non-Euclidean Geometry “15-Years of Lobachevskii Geometry” [in Russian], Kazan, 1976, Moscow (1977), pp. 133–141.
V. R. Kaigorodov, “Semisymmetric Lorentzian spaces with perfect holonomy group,” in: Gravitation and Relativity Theory [in Russian], Vols. 14–15, Kazan Univ. (1978), pp. 113–120.
V. R. Kaigorodov, “On gravitational fields with recurrent complex-conformal curvature,” in: Gravitation and Relativity Theory [in Russian], Vols. 14–15, Kazan Univ. (1978), pp. 180–185.
V. R. Kaigorodov and A. B. Pestov, “Constant vector fields in Einstein spaces,” in: Gravitation and Relativity Theory [in Russian], Vol. 6, Kazan Univ. (1969), pp. 46–51.
E. Cartan, Geometry of Lie Groups and Symmetric Spaces [Russian translation], IL, Moscow (1949).
A. Lichnerowicz, Théorie Globale des Connexions et des Groupes d'Holonomie, Ed. Cremonese, Rome (1955).
A. Z. Petrov, New Methods in General Relativity Theory [in Russian], Nauka, Moscow (1966).
P. A. Shirokov, “Constant vector fields and tensor fields of 2nd order in Riemannian spaces,” Izv. Fiz. Mat. O-va KGU, Ser. 2,25, 86–114 (1925).
P. A. Shirokov, “Symmetric conformally Euclidean spaces,” Izv. Fiz. Mat. O-va KGU, Ser. 3,11, 9–27 (1938).
P. A. Shirokov, “Symmetric spaces of first class,” in: Collected Works on Geometry [in Russian], Kazan Univ. (1966), pp. 366–383.
K. Yano and S. Bochner, Curvature and Betti Numbers, Princeton Univ. Press, Princeton, N.J. (1953).
T. Adati and T. Miyasawa, “On conformally symmetry spaces,” Tensor,18, No. 3, 335–342 (1967).
T. Adati and T. Miyasawa, “On Riemannian spaces with recurrent conformal curvature,” Tensor,18, No. 3, 348–354 (1967).
T. Adati and S. Yamaguchi, “On some transformations in Riemannian recurrent spaces,” TRU Math.,3, 8–12 (1967).
L. R. Ahuja and R. Behari, “H-Projective-symmetric and recurrent Kähler spaces,” Tensor,26, No. 1, 102–104 (1972).
M. Cahen and R. McLenaghan, “Metriques des espaces lorentziens symmétriques à quatre dimensions,” C. R. Acad. Sci. Paris,266, No.2, A1125-A1128 (1968).
M. C. Chaki, “On conformally recurrent Kähler spaces,” Tensor,25, No. 2, 179–182 (1972).
M. C. Chaki and R. A. N. Chowdhury, “On conformally recurrent spaces of second order,” J. Austral. Math. Soc.,10, Nos. 1–2, 155–161 (1969).
R. A. N. Chowdhury, “Some theorems on recurrent spaces of second order,” Bull. Acad. Polon. Sci. Math. Astron. Phys.,15, No. 3, 171–176 (1967).
R. A. N. Chowdhury, “On recurrent spaces of second order,” Bull. Acad. Polon. Sci. Ser. Sci. Math. Astron. Phys.,16, No. 12, 955–959 (1968).
A. Cochina, “Affine spaces with recurrent curvature tensor,” Stud. si Cere. Mat.,23, No. 8, 1187–1191 (1971).
D. K. Datta, “Some theorems on symmetric recurrent tensors of the second order,” Tensor,15, No. 1, 61–65 (1964).
D. K. Datta, “Notes on Ricci-recurrent spaces,” Rend. Circ. Mat. Palermo,14, No. 3, 281–286 (1965).
D. K. Datta, “On the recurrence covector,” Rev. Roum. Math. Pures Appl.,17, No. 2, 229–235 (1972).
P. Enghis, “Sur les espaces Vn recurrents et Ricci recurrents,” Stud. Univ. Babes-Bolyai Ser. Math. Mech.,17, No. 1, 3–6 (1972).
E. Glodek, “On Riemannian conformally symmetric spaces admitting projective collineations,” Colloq. Math.,26, 123–128 (1972).
E. Glodek and W. Roter, “On torse-forming vector fields in Ricci-recurrent and secondorder recurrent spaces,” Fasc. Math., No. 4, 15–24 (1969).
B. Gupta, “On projective-symmetric spaces,” J. Aust. Math. Soc.,4, No. 1, 113–121 (1964).
V. Hlavaty, “Rigid motion in a Riemannian space Vn. I. A recurrent Vn,” Rend. Circ. Mat. Palermo,2, No. 9, 51–77 (1962).
G. R. Kumar, “On recurrent spaces of second order,” Bull. Cl. Sci. Acad. R. Belg.,58, No. 4, 481–492 (1972).
R. A. Kumar, “On generalized 2-recurrent tensors in Riemannian spaces,” Bull. Cl. Sci. Acad. R. Belg.,58, No. 2, 220–228 (1972).
K. B. Lal and S. S. Singh, “On Kählerian spaces with recurrent Bochner curvature,” Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis., Nat. Natur.,51, Nos. 3–4, 213–220 (1971).
Y. D. Liang, “Characterizations of certain K*-space,” Chinese Math.,9, No. 3, 383–389 (1967).
A. Lichnerowicz, “Courbure, nombres de Betti et espaces symmétriques,” in: Proc. Int. Cong. Math., Vol. 2, 216–233 (1952).
M. L. Marziale, “Caratteristica di Eulero-Poincare delle varieta ricorrenti,” Rend. Math.,4, No. 2, 295–301 (1971).
S. B. Mathur, “On recurrent spaces,” Proc. Nat. Inst. Sci. India,A34, No. 1, 40–49 (1968).
M. Matsumoto, “On Riemannian spaces with recurrent protective curvature,” Tensor,19, No. 1, 11–18 (1968).
M. Matsumoto, “On h-isotropic and Ch-recurrent Finsler spaces,” J. Math. Kyoto Univ.,11, No. 1, 1–9 (1971).
R. G. Mc. Lenaghan and J. Leroy, “Complex recurrent space-times,” Proc. R. Soc. London,A327, No. 1569, 229–249 (1972).
R. G. Mc. Lenaghan and A. H. Thompson, “Second-order recurrent space-times in general relativity,” Lett. Nuovo Cimento,5, No. 7, 563–564 (1972).
R. S. Mishra and G. P. Pokhariyal, “Curvature tensors in Riemannian manifolds,” Indian J. Pure Appl. Math.,2, No. 3, 529–532 (1971).
R. B. Misra, “A symmetric Finsler space,” Tensor,24, 346–350 (1972).
T. Miyasawa, “On conformal transformations of Riemannian spaces with recurrent conformal curvature,” TRU Math.,3, 19–24 (1964).
J. Mogi, “A remark on recurrent curvature spaces,” Kodai Math. Sem. Rep. (1950).
A. Moor, “Räume von zweifach rekurrenter Krümmung,” Publs. Math.,18, Nos. 1–4, 165–169 (1971).
A. Moor, “Untersuchungen in Räumen mit rekurrenter Krümmung,” J. Reine Angew. Math.,199, Nos. 1–2, 91–99 (1958).
A. Moor, “Über Finslerraume von zweifach rekurrenter Krümmung,” Acta Math. Acad. Sci. Hung.,22, Nos. 3–4, 453–465 (1972).
A. Moor, “Über die Konstruktion von metrisch-affinen Räumen von rekurrenter Krummung,” Colloq. Math.,26, 109–113 (1972).
M. Nagaraj, “On some conformal properties of recurrent Riemannian spaces,” Tensor,19, No. 1, 19–26 (1968).
J. Navez, “Riemannian spaces with second-order recurrent tensor of conformal curvature,” Bull. Soc. R. Sci. Liege,40, Nos. 3–4, 110–115 (1971).
K. Nomizu, “On hypersurfaces satisfying a certain condition on the curvature tensors,” Tohoku Math. J.,20, No. 1, 46–59 (1968).
F. Oktem, “Are the neutrinos an aspect of Riemannian geometry?,” Nuovo Cimento,A1, No. 1, 38–48 (1971).
N. Prakash, “Some remarks on recurrent and Ricci-recurrent spaces,” Ganita,12, No. 2, 105–114 (1961).
B. N. Prasad, “On a Kahlerian space with recurrent Bochner curvature tensor,” Atti Acad. Naz. Lincei. Rend. Cl. Sci. Fis., Mat. e Natur,53, Nos. 1–2, 87–93 (1972).
M. Prvanovitch, “Projective and conformal transformations in recurrent and Ricci-recurrent Riemannian spaces,” Tensor,12, No. 3, 219–226 (1962).
M. Prvanovitch, “Poludekomponovani rekurentni rimanovi prostori,” Godishn. Filoz. Fak. Novom Sadu,11, No. 2, 717–720 (1968).
W. Roter, “Some remarks on second order recurrent spaces,” Bull. Acad. Polon. Sci. Ser. Sci. Math., Astron. Phys.,12, No. 4, 207–211 (1964).
W. Roter, “A note on second order recurrent spaces,” Bull. Acad. Polon. Sci. Ser. Sci. Math., Astron. Phys.,12, No. 10, 621–626 (1964).
W. Roter, “On conformally symmetric 2-Ricci-recurrent spaces,” Colloq. Math.,26, No. 1, 116–122 (1972).
W. Roter, “Some remarks on infinitesimal projective transformations in recurrent and Ricci-recurrent spaces,” Colloq. Math.,15, No. 1, 121–127 (1966).
H. S. Ruse, “On simply harmonic spaces,” J. Lond. Math. Soc.,21, Part 4 (1946).
H. S. Ruse, “Three-dimensional spaces of recurrent curvature,” Proc. Lond. Math. Soc. (Ser. 2),50, Part 6 (1949).
H. S. Ruse, “On simply harmonic ‘kappa-spaces’ of four dimensions,” Proc. Lond. Math. Soc. (Ser. 2),50, 317–329 (1948).
D. W. Sciama, “Recurrent radiation in general relativity,” Proc. Cambridge Philos. Soc.,57, No. 2, 436–439 (1961).
R. N. Sen, “Finsler spaces of recurrent curvature,” Tensor,19, No. 3, 291–299 (1968).
S. P. Singh, “A special Kawaguchi space of recurrent curvature tensor field of second order,” Tensor,23, No. 2, 174–178 (1972).
F. Söler, “Sur les varietes recurrentes conformement planes,” C. R. Acad. Sci.,277, No. 13, A601-A603 (1973).
K. Takano, “Decomposition of curvature tensor in a recurrent space,” Tensor,18, No. 3, 343–347 (1967).
K. Takano, “On a space with bi-recurrent curvature,” Tensor,22, No. 3, 329–332 (1971).
K. Takano, “Some remarks on the affine motions in a space with recurrent curvature,” Tensor,12, No. 1, 84–89 (1962).
K. Takano, “On the existence of affine motions in a space with recurrent curvature. I,” Tensor,17, No. 1, 68–73 (1966).
K. Takano, “On the existence of affine motions in a space with recurrent curvature. II,” Tensor,17, No. 2, 212–216 (1966).
H. Takeno, “On the n-recurrency of spherical symmetric space-times. I,” Res. Inst. Theor. Phys. Hiroshima Univ., No. RRK-5 (1970).
H. Takeno, “On the n-recurrency of spherical symmetric space-times. II,” Res. Inst. Theor. Phys. Hiroshima Univ., No. RRK-6 (1970).
H. Takeno, “On the n-recurrency of spherical symmetric space-times. III,” Res. Inst. Theor. Phys. Hiroshima Univ., No. RRK-9 (1970).
H. Takeno, “On the n-recurrency of H-spaces in general relativity,” Res. Inst. Theor. Phys. Hiroshima Univ., No. RRK-11 (1970).
H. Takeno, “On n-recurrent H space-times,” Tensor,22, No. 3, 374–378 (1971).
H. Takeno, “On the recurrency of the space-times. V,” Res. Inst. Theor. Phys. Hiroshima Univ., No. RRK-3 (1973).
A. H. Thompson, “On conformally flat 2-recurrent Riemannian spaces,” Q. J. Math.,20, No. 80, 505–510 (1969).
A. H. Thompson, “On second-order recurrent space-times. I,” Bull. Acad. Polon. Sci. Ser. Sci. Math., Astron. Phys.,16, No. 2, 121–124 (1968).
A. H. Thompson, “On second-order recurrent space-times. II,” Bull. Acad. Polon. Sci. Ser. Sci. Math., Astron. Phys.,17, No. 10, 661–670 (1969).
A. H. Thompson, “On Riemannian spaces of second-order recurrent curvature,” Bull. Acad. Polon. Sci. Ser. Sci. Math., Astron. Phys.,18, No. 6, 335–340 (1970).
G. G. Vrănceanu, “Asupra spatiilor cu tensor de curbură recurent,” Stud. si Cerc. Mat.,22, No. 6, 947–953 (1970).
A. G. Walker, “On Ruse's spaces of recurrent curvature,” Proc. Lond. Math. Soc.,51, 36–64 (1950).
A. G. Walker, “Symmetric harmonic spaces,” J. Lond. Math. Soc.,21, No. 81 1946).
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Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 14, pp. 177–204, 1983.
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Kaigorodov, V.R. Structure of space-time curvature. J Math Sci 28, 256–273 (1985). https://doi.org/10.1007/BF02105213
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DOI: https://doi.org/10.1007/BF02105213