ON SPECTRA OF SOME TENSORS OF SIX-DIMENSINAL KÄHLERIAN AND NEARLY-KÄHLERIAN SUBMANIFOLDS OF CAYLEY ALGEBRA
Article Sidebar
Main Article Content
Abstract
Six-dimensional Kählerian and nearly-Kählerian submanifolds of Cayley algebra are considered. Spectra of some classical tensors of such submanifolds of the octave algebra are computed. It is proved that a nearly-Kählerian six-dimensional submanifold of Cayley algebra is conharmonically flat if and only if it is holomorphically isometric to the complex Euclidean space with a canonical Kählerian structure.
Article Details
This work is licensed under a Creative Commons Attribution 4.0 International License.
Tikrit Journal of Pure Science is licensed under the Creative Commons Attribution 4.0 International License, which allows users to copy, create extracts, abstracts, and new works from the article, alter and revise the article, and make commercial use of the article (including reuse and/or resale of the article by commercial entities), provided the user gives appropriate credit (with a link to the formal publication through the relevant DOI), provides a link to the license, indicates if changes were made, and the licensor is not represented as endorsing the use made of the work. The authors hold the copyright for their published work on the Tikrit J. Pure Sci. website, while Tikrit J. Pure Sci. is responsible for appreciate citation of their work, which is released under CC-BY-4.0, enabling the unrestricted use, distribution, and reproduction of an article in any medium, provided that the original work is properly cited.
References
[1] Abu-Saleem A., Banaru M. Some applications of
Kirichenko tensors. An. Univ. Oradea, 17 (2010), N5,
201-208.
[2] Abu-Saleem A., Banaru M. On parakählerian and
C-parakählerian manifolds. Abhath Al-Yarmouk.
Basic Sci.&Eng. Irbid, 15 (2006),.N1, 101-109.
[3] Banaru M.B. Hermitian geometry of sixdimensional
submanifolds of Cayley algebra (PhD
thesis). MSPU, Moscow, 1993 (in Russian).
[4] Banaru M.B. Two theorems on cosymplectic
hypersurfaces of six-dimensional Hermitian
submanifolds of Cayley algebra. Journal of Harbin
Institute of Technology, 8 (2000), N 1, 38-40.
[5] Banaru M.B. A note on 6-dimensional Hermitian
submanifolds of Cayley algebra. Buletinul Stintific
Univ. Timisoara. V. 45 (59) (2000), N2, 17-20.
[6] Banaru M.B. On tensors spectra of sixdimensional
Hermitian submanifolds of Cayley
algebra. Recent Problems in Field Theory, Kazan, 1
(2000), 18-21.
[7] Banaru M.B. A note on R2-and cR2-manifolds.
Journal of Harbin Institute of Technology, 9 (2002),
N2, 136-138.
[8] Banaru M.B. A note on six-dimensional G1-
submanifolds of octave algebra. Taiwanese Journal of
Mathematics, 6 (2002), N3, 383-388.
[9] Banaru M.B. Hermitian manifolds and Ucosymplectic
hypersurfaces axiom. Journal of
Sichuan Normal University (Natural Science), 26
(2003), N3, 261-263.
[10] Calabi E. Construction and properties of some 6-
dimensional almost complex manifolds. Trans. Amer.
Math. Soc., .87 (1958), N2, 407-438.
[11] Cartan E. Riemannian geometry in an orthogonal
frame. MGU, Moscow, 1960.
[12] Ejiri N. Totally real submanifolds in a 6-sphere.
Proc. Amer. Math. Soc., 83 (1981), 759-763.
[13] Funabashi S., Pak J.S. Tubular hypersurfaces of
the nearly Kähler 6-sphere. Saitama Math. J.,19
(2001), 13-36.
[14] Gray A. Some examples of almost Hermitian
manifolds. Illinois J. Math., 10 (1966), N2, 353-366.
[15] Gray A. Six-dimensional almost complex
manifolds defined by means of three-fold vector cross
products. Tôhoku Math. J., 21 (1969), N4, 614-620.
[16] Gray A. Vector cross products on manifolds.
Trans. Amer. Math. Soc.,141 (1969), .465-504.
[17] Gray A. Almost complex submanifolds of the six
sphere. Proc. Amer. Math. Soc., 20, (1969), 277-280.
[18] Gray A. Nearly Kähler manifolds. J. Diff.
Geom., 4 (1970), 283-309.
[19] Gray A. The structure of nearly Kähler
manifolds. Math. Ann., 223 (1976), 223-248.
[20] Gray A., Hervella L.M. The sixteen classes of
almost Hermitian manifolds and their linear
invariants. Ann. Mat., Pura Appl., 123 (1980), N4,
35-58.
[21] Haizhong Li. The Ricci curvature of totally real
3-dimensional submanifolds of the nearly-Kaehler 6-
sphere. Bull. Belg. Math. Soc.-Simon Stevin, 3
(1996), 193-199.
[22] Haizhong Li., Gouxin Wei. Classification of
Lagrangian Willmore submanifolds of the nearly-
Kaehler 6-sphere (1) 6 S with constant scalar
curvature. Glasgow Math. J., 48 (2006), 53-64.
[23] Hashimoto H. Characteristic classes of oriented
6-dimensional submanifolds in the octonions. Kodai
Math. J., 16 (1993), 65-73.
[24] Hashimoto H. Oriented 6-dimensional
submanifolds in the octonions. Internat. J. Math.
Math. Sci.,18 (1995), 111-120.
[25] Hashimoto H., Koda T., Mashimo K., Sekigawa
K., Extrinsic homogeneous Hermitian 6-dimensional
submanifolds in the octonions, Kodai Math. J., 30
(2007), 297-321.
[26] Ishii Y. On conharmonic transformations,
Tensor. N.S., 7 (1957), 73-80.
[27] Kirichenko V.F. On nearly-Kählerian structures
induced by means of 3-vector cross products on sixdimensional
submanifolds of Cayley algebra, Vestnik
MGU, 3 (1973), 70-75.
[28] Kirichenko V.F. Differential geometry of Kspaces.
Problems of Geometry, 8 (1977), 139-161.
[29] Kirichenko V.F. Classification of Kählerian structures, defined by means of three-fold vector cross products on six-dimensional submanifolds of Cayley algebra, Izvestia Vuzov. Math., 8(1980), 32-38.
[30] Kirichenko V.F. Rigidity of almost Hermitian structures induced by 3-vector cross products on six-dimensional submanifolds of Cayley algebra. Ukrain Geom. Sbornik, 25 (1982), 60-68.
[31] Kirichenko V.F. Methods of generalized Hermitian geometry in the theory of almost contact metric manifolds. Problems of Geometry, 18 (1986), 25-71.
[32] Kirichenko,V.F.: Hermitian geometry of six–dimensional symmetric submanifolds of Cayley algebra. Vestnik MGU, 3(1994), 6-13.
[33] Kirichenko V.F. Generalized quasi-Kählerian manifolds and axioms of CR-submanifolds in generalized Hermitian geometry, I. Geom. Dedicata, 51 (2004), 75-104.
[34] Kirichenko V.F. Generalized quasi-Kählerian manifolds and axioms of CR-submanifolds in
generalized Hermitian geometry, II. Geom. Dedicata, 52 (2004), 53-85.
[35] Kirichenko V.F. Differential - geometrical structures on manifolds. MSPU, Moscow, 2003 (in Russian).
[36] KirichenkoV.F., Shihab A.A. On geometry of the tensor of conharmonic curvature of nearly-Kählerian manifolds. Fundamental and Applied Mathematics, 16 (2010), N2, 43-54.
[37] Koszul J. Varietes Kählerienes. Notes, Sao-Paolo, 1957.
[38] Sekigawa K. Almost complex submanifolds of a six-dimensional sphere, Kodai Math. J., .6 (1983), 174-185.
[39] Shihab A.A. Geometry of the tensor of conharmonic curvature of nearly-Kählerian manifolds (PhD thesis). Moscow State Pedagogical Univ., 2011 (in Russian).
[40] Vranchen L. Special Lagrangian submanifolds of the nearly Kaehler 6-sphere. Glasgow Math. J., 45 (2003), 415-426.