×

Pseudo-Killing and pseudoharmonic vector fields on a Riemann-Cartan manifold. (English. Russian original) Zbl 1197.53049

Math. Notes 87, No. 2, 248-257 (2010); translation from Mat. Zametki 87, No. 2, 267-279 (2010).
Summary: Six classes of Riemann-Cartan manifolds are distinguished in an invariant way. Geometric characteristics of some of the distinguished classes of Riemann-Cartan manifolds are found, and also conditions hindering the existence, are determined. The local geometry of Riemann-Cartan manifolds carrying pseudo-Killing and pseudoharmonic vector fields is studied. Conditions hindering the existence “in the large” of pseudo-Killing and pseudoharmonic vector fields on Riemann-Cartan manifolds are obtained.

MSC:

53C20 Global Riemannian geometry, including pinching
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] É. Cartan, ”Sur les variétés à connexion affine et la théorie de la relativité généralisée (premiére partie),” Ann. Sci.école Norm. Sup. (3) 40, 325–412 (1923).
[2] É. Cartan, ”Sur les variétés à connexion affine et la théorie de la relativité généralisée (premiére partie) (suite),” Ann. Sci.école Norm. Sup. (3) 41, 1–25 (1924).
[3] É. Cartan, ”Sur les variétés à connexion affine et la théorie de la relativité généralisée (deuxiéme partie),” Ann. Sci.école Norm. Sup. (3) 42, 17–88 (1925). · JFM 51.0582.01
[4] R. Penrose, ”Spinors and torsion in general relativity,” Found. Phys. 13(3), 325–339 (1983).
[5] A. Trautman, ”The Einstein-Cartan theory,” in Encyclopedia of Mathematical Physics, (Elsevier, Oxford, 2006), Vol. 2, pp. 189–195; arXiv: gr-qc/0606062.
[6] D. Puetzfeld, ”Prospects of non-Riemannian cosmology,” in Proceedings of 22nd Texas Symposium on Relativistic Astrophysics at Stanford University, December 13–17, 2004, paper no. 1221 (Stanford Univ. Press, California, 2004); arXiv: astro-ph/0501231.
[7] R. T. Hammond, ”Torsion gravity,” Rep. Progr. Phys. 65(5), 599–649 (2002).
[8] K. Yano, ”On semi-symmetric metric connection,” Rev. Roumaine Math. Pures Appl. 15, 1579–1586 (1970). · Zbl 0213.48401
[9] Z. Nakao, ”Submanifolds of a Riemannian manifold semisymmetric metric connections,” Proc. Amer. Math. Soc. 54(1), 261–266 (1976). · Zbl 0323.53040
[10] B. Barua and A. K. Ray, ”Some properties of semisymmetric metric connection in a Riemannian manifold,” Indian J. Pure Appl.Math. 16(7), 736–740 (1985). · Zbl 0577.53017
[11] J. Sengupta, U. C. De, and T. Q. Binh, ”On a type of semi-symmetric non-metric connection on a Riemannian manifold,” Indian J. Pure Appl.Math. 31(12), 1659–1670 (2000). · Zbl 0980.53035
[12] G. Muniraja, ”Manifolds admitting a semi-symmetric metric connection and a generalization of Schur’s theorem,” Int. J. Contemp.Math. Sci. 3, 1223–1232 (2008). · Zbl 1169.53329
[13] S. Bochner and K. Yano, ”Tensor-fields in non-symmetric connections,” Ann. of Math. (2) 56(3), 504–519 (1952). · Zbl 0048.15802
[14] K. Yano and S. Bochner, Curvature and Betti Numbers (Princeton University Press, Princeton, N. J., 1953; Inostr. Lit., Moscow, 1957).
[15] S. I. Goldberg, ”On pseudo-harmonic and pseudo-Killing vectors in metric manifolds with torsion,” Ann. of Math. (2) 64(2), 364–373 (1956). · Zbl 0073.38902
[16] Y. Kubo, ”Vector fields in a metric manifold with torsion and boundary,” Kōdai Math. Sem. Rep. 24(4), 383–395 (1972). · Zbl 0259.53038
[17] J. A. Schouten, Ricci-Calculus. An Introduction to Tensor Analysis and its Geometrical Applications, in Grundlehren Math. Wiss. (Springer-Verlag, Berlin, 1954), Vol. 10.
[18] S. E. Stepanov, ”On an analytic method in general relativity,” Teoret. Mat. Fiz. 122(3), 482–496 (2000) [Theoret. and Math. Phys. 122 (3), 402–414 (2000)]. · Zbl 0971.83009
[19] I. A. Gordeeva, ”On the classification of nonsymmetric metric connections,” in Collection of Works of the G. F. Laptev International Geometric Seminar (Penz. Gos. Ped. Inst., Penza, 2006), pp. 30–37 [in Russian].
[20] I. A. Gordeeva, ”Pseudo-Killing vector fields on Riemann–Cartan manifolds,” in Abstracts of Reports of the International Mathematical Conference ”Geometry in Odessa-2008”, Odessa, May 19–24, 2008 (Fond ”Nauka,” Odessa, 2008), pp. 73–75 [in Russian].
[21] I. A. Gordeeva and S. E. Stepanov, ”Vanishing theorem for pseudoharmonic vector fields on a Riemann–Cartan manifold,” in Abstracts of Reports of the International Conference on Differential Equations and Dynamical Systems, Suzdal, June 27–July 2, 2008 (VlGU, Vladimir, 2008), pp. 71–73 [in Russian].
[22] L. Berard-Bergery, M. Berger, and C. Houzel, Géométrie riemannienne en dimension 4, Seminar: Arthur Besse, Riemannian geometry in dimension 4; Paris, 1978/1979 (CEDIC, Paris, 1981; Mir, Moscow, 1985).
[23] S. E. Stepanov, ”On conformal Killing 2-form of the electromagnetic field,” J. Geom. Phys. 33(3–4), 191–209 (2000). · Zbl 0977.53013
[24] A. P. Norden, Spaces with Affine Connection (Nauka, Moscow, 1976) [in Russian]. · Zbl 0925.53007
[25] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. 2 (Interscience Publishers, a division of John Wiley & Sons, New York-London-Sydney, 1969; Nauka, Moscow, 1981). · Zbl 0175.48504
[26] V. F. Kirichenko, ”Methods of generalized Hermitian geometry in the theory of almost contact manifolds,” in Problems in Geometry [in Russian] Vol. 18, Itogi Nauki i Tekhniki [Progress in Science and Technology], Vsesoyuz. Inst. Nauchn. i Tekhn. Inform. (VINITI), Moscow, 1986, pp. 25–71.
[27] A. Gray and L. M. Hervella, ”The sixteen classes of almost Hermitian manifolds and their linear invariants,” Ann.Mat. Pura Appl. (4) 123(1), 35–58 (1980). · Zbl 0444.53032
[28] L. Friedland and Ch.-Ch. Hsiung, ”A certain class of almost Hermitian manifolds,” Tensor (N. S.) 48(3), 252–263 (1989). · Zbl 0718.53027
[29] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. 1 (Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963; Nauka, Moscow, 1981). · Zbl 0119.37502
[30] S. Tanno, ”Partially conformal transformations with respect to (m - 1)-dimensional distributions of m-dimensional Riemannian manifolds,” Tôhoku Math. J. (2) 17(4), 358–409 (1965). · Zbl 0132.16101
[31] A. V. Dubinkin and A. P. Shirokov, ”On the question of infinitesimal generalized-conformal transformations,” in Trudy Geom. Sem. Kazan. Univ. (Izd. Kazan. Univ., Kazan, 1983), Vol. 15, pp. 26–34 [in Russian]. · Zbl 0555.53011
[32] B. L. Reinhart, Differential Geometry of Foliations. The Fundamental Integrability Problem, in Ergeb. Math. Grenzgeb. (Springer-Verlag, Berlin, 1983), Vol. 99. · Zbl 0506.53018
[33] S. E. Stepanov, ”An integral formula for a Riemannian almost-product manifold,” Tensor (N. S.) 55(3), 209–214 (1994). · Zbl 0831.53023
[34] L. P. Eisenhart, Riemannian Geometry (Princeton Univ. Press, Princeton, NJ, 1926; Inostr. Lit., Moscow, 1948).
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.