# zbMATH — the first resource for mathematics

Geometry of 6-dimensional Hermitian manifolds of the octave algebra. (English. Russian original) Zbl 1365.53064
J. Math. Sci., New York 207, No. 3, 354-388 (2015); translation from Itogi Nauki Tekh., Ser. Sovrem. Mat. Prilozh., Temat. Obz. 126 (2013).
This paper is a very useful survey of recent results in the geometry of 6-dimensional Hermitian manifolds of Cayley algebras. The author describes Gray-Hervella classes of almost Hermitian structures on 6-dimensional oriented submanifolds of Cayley algebras and proposes a characterization of these in terms of the Kirichenko tensors and the configuration tensor.
The author is a major contributor to this field, more than one third of the 188 References belong to himself. The article is divided into four sections:
1. Almost Hermitian structures on 6-dimensional oriented submanifolds of Cayley algebras.
2. Gray-Hervella classes of almost Hermitian structures on 6-dimensional oriented submanifolds of Cayley algebras.
3. Hermitian structures on 6-dimensional oriented submanifolds of Cayley algebras.
4. Approximately Kählerian structures on 6-dimensional oriented submanifolds of Cayley algebras.

##### MSC:
 53C55 Global differential geometry of Hermitian and Kählerian manifolds 53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
Full Text:
##### References:
 [1] A. Abu-Saleem, “Some remarks on almost Hermitian manifolds with $$J$$-invariant Ricci tensor,” Int. Math. Forum, 5, No. 2 (2010). · Zbl 1193.53056 [2] Abu-Saleem, A; Banaru, M, Some applications of kirichenko tensors, An. Univ. Oradea, Fasc. Mat., 17, 201-208, (2010) · Zbl 1212.53045 [3] Arsen’eva, OE; Kirichenko, VF, Self-dual geometry of generalized Hermitian surfaces, Mat. Sb., 189, 21-44, (1998) · Zbl 0907.53023 [4] M. B. Banaru, A new characterization of the classes of almost Hermitian Gray-Hervella structures [in Russian], preprint, Smolensk. State Pedagogical Institute (1992). Deposited at the All-Russian Institute for Scientific and Technical Information (VINITI), Moscow, No. 3334-B92. [5] M. B. Banaru, “Gray-Hervella classes of almost Hermitian structures on 6-dimensional submanifolds of Cayley algebras,” in: Proc. Int. Mat. Conf. Dedicated to the 200th Anniversary of N. I. Lobachevskii, Akad. Nauk Resp. Belarus’, 1, Minsk (1993), p. 40. · Zbl 0182.24603 [6] M. B. Banaru, On the Gray-Hervella classification of almost Hermitian structures on 6-dimensional submanifolds of Cayley algebras [in Russian], Smolensk. State Pedagogical Institute (1993). Deposited at the All-Russian Institute for Scientific and Technical Information (VINITI), Moscow, No. 118-B93. · Zbl 0080.37601 [7] M. B. Banaru, On almost Hermitian structures induced by 3-vector products on 6-dimensional submanifolds of Cayley algebras [in Russian], Smolensk. State Pedagogical Institute (1993). Deposited at the All-Russian Institute for Scientific and Technical Information (VINITI), Moscow, No. 1282-B93. · Zbl 0484.53014 [8] M. B. Banaru, “On the minimality of almost Hermitian 6-dimensional submanifolds of Cayley algebras,“ in Proc. Int. Sci. Conf. “Pontryagin Readings-IV,” Voronezh (1993), p. 17. · Zbl 1076.53034 [9] M. B. Banaru, “On the para-Kählerian property of 6-dimensional Hermitian submanifolds of Cayley algebras,” in: Differential Geometry of Manifolds of Figures [in Russian], 25, Kaliningrad State Univ., Kaliningrad (1994), pp. 15-18. [10] M. B. Banaru, “On the para-Kählerian property of six-dimensional Hermitian submanifolds of Cayley algebras,” in: Webs and Quasigroups [in Russian], Kalinin, (1994), pp. 81-83. · Zbl 0896.53044 [11] M. B. Banaru, “Gray-Hervella classes of almost Hermitian structures on 6-dimensional submanifolds of Cayley algebras,” in: Proc. Moscow State Pedagogical Univ., Moscow (1994), pp. 36-38. · Zbl 0526.53055 [12] M. B. Banaru, “On the holomorphic bisectional curvature of 6-dimensional Hermitian submanifolds of Cayley algebras,” in: Differential Geometry of Manifolds of Figures [in Russian], 28, Kaliningrad State Univ., Kaliningrad (1997), pp. 7-9. · Zbl 0888.53037 [13] M. B. Banaru, “On almost Hermitian structures induced by 3-vector products on 6-dimensional submanifolds of the octave algebra,” in: Polyanalytic Functions: Boundary Properties and Boundary-Value Problems [in Russian], Smolensk (1997), pp. 113-117. · Zbl 1138.32013 [14] M. B. Banaru, “On the properties of the curvature of 6-dimensional Hermitian submanifolds of Cayley algebras,“ in: Proc. Int. Semin. “Selected Questions of Higher Mathematics and Informatics”, Smolensk (1997), pp. 25-26. · Zbl 1396.53070 [15] M. B. Banaru, “On spectra of the most important tensors of 6-dimensional Hermitian submanifolds of Cayley algebras,” in: The Newest Problems of the Field Theory [in Russian], Kazan’ (2000), pp. 18-22. · Zbl 1029.53038 [16] M. B. Banaru, “On 6-dimensional submanifolds of Cayley algebras,” in: Differential Geometry of Manifolds of Figures [in Russian], 31, Kaliningrad State Univ., Kaliningrad (2000), pp. 6-8. · Zbl 1064.53013 [17] M. B. Banaru, “On 6-dimensional $$G$$_{1}-submanifolds of the octave algebra,” in: Proc. Moscow State Pedagogical Univ., Moscow (2000), pp. 165-171. [18] M. B. Banaru, “A note on six-dimensional Vaisman-Gray submanifolds of Cayley algebras,” in: Webs and Quasigroups [in Russian], Tver (2000), pp. 139-142. · Zbl 1007.53051 [19] M. B. Banaru, “A note on six-dimensional Hermitian submanifolds of Cayley algebras,” Bul. S¸tiint¸. Univ. Politeh. Timi¸s., 45(59), No. 2, 17-20 (2000). · Zbl 1052.53046 [20] Banaru, MB, Six theorems on six-dimensional Hermitian submanifolds of Cayley algebras, Izv. Akad. Nauk Resp. Moldova. Ser. Mat., 3, 3-10, (2000) · Zbl 1031.53087 [21] Banaru, MB, On six-dimensional Hermitian submanifolds of Cayley algebras satisfying the $$g$$-cosymplectic hypersurfaces axiom, Ann. Univ. Sofia Fac. Math. Inform., 94, 91-96, (2000) · Zbl 1079.53506 [22] M. B. Banaru, “On 6-dimensional $$W$$_{4}-submanifolds of the octave algebra,” in: Proc. Moscow State Pedagogical Univ., Moscow (2001), pp. 46-48. · Zbl 1099.53049 [23] M. B. Banaru, “A new example of an $$R$$_{2}-manifold,“ in: Proc. VIII Int. Sci. Conf. “Mathematics. Computer. Education,” Moscow (2001), p. 125. [24] M. B. Banaru, “On conformally flat $$c$$-para-Kählerian manifolds,“ in: Proc. IX Int. Conf. “Mathematica. Education. Economics. Ecology,” Cheboksary (2001), p. 35. · Zbl 1080.53048 [25] M. B. Banaru, “On $$R$$_{2}- and cR_{2}-manifolds,” in: Mathematics. Computer. Education [in Russian], 8, Moscow (2001), pp. 471-476. · Zbl 0937.53033 [26] M. B. Banaru, “On locally Euclidean para-Kählerian manifolds,” in: Proc. Ukrain. Math. Congr., Kiev (2001), pp. 13-14. · Zbl 1041.53034 [27] M. B. Banaru, “On AH-manifolds with $$J$$-invariant Ricci tensor,” in: Proc. IV Int. Conf. on Geometry and Topology, Cherkassy (2001), p. 9. · Zbl 1036.53019 [28] M. B. Banaru, “On para-Kählerian and $$c$$-para-Kählerian manifolds,” in: Differential Geometry of Manifolds of Figures [in Russian], 32, Kaliningrad State Univ., Kaliningrad (2001), pp. 8-13. · Zbl 1054.53031 [29] M. B. Banaru, “On the geometry of cosymplectic hyperplanes of 6-dimensional Hermitian submanifolds of Cayley algebras,“ in: Proc. Int. Sci. Conf. “Volga-2001” (Petrovskii Readings), Kazan’ (2001), p. 25. · Zbl 1041.53034 [30] M. B. Banaru, “On the Einstein property of 6-dimensional Hermitian submanifolds of Cayley algebras,” in: Studies in the Boundary-Value Problems of Complex Analysis and Differential Equations [in Russian], 3, Smolensk (2001), pp. 28-35. · Zbl 0236.53048 [31] M. B. Banaru, “On the geometry of 6-dimensional Hermitian submanifolds of Cayley algebras,” in: Invariant Methods of the Study of Structures on Manifolds in Geometry, Analysis, and Mathematical Physics [in Russian] (L. E. Evtushik and A. K. Rybnikov, eds.), 1, Moscow (2001), pp. 16-20. [32] Banaru, MB, On six-dimensional Hermitian submanifolds of Cayley algebras, Stud. Univ. Babeş-Bolyai. Math., 46, 11-14, (2001) · Zbl 1027.53029 [33] Banaru, MB, Two theorems on cosymplectic hypersurfaces of six-dimensional Hermitian submanifolds of Cayley algebras, J. Harbin Inst. Tech., 8, 38-40, (2001) [34] M. B. Banaru, “A note on Kirichenko tensors,“ in: Proc. Int. Sci. Conf. “Volga-2001” (Petrovskii Readings), Kazan’ (2001), p. 26. · Zbl 0263.53019 [35] M. B. Banaru, “A new characterization of the Gray—Hervella classes of almost Hermitian manifolds,” in: Proc. 8th Int. Conf. on Differential Geometry and Its Aplications, Opava, Czech Republic (2001), p. 4. [36] Banaru, MB, A note on RK- and CRK-manifolds, Izv. Akad. Nauk Resp. Moldova. Ser. Mat., 1, 37-43, (2001) · Zbl 1036.53018 [37] Banaru, MB, On six-dimensional G1-submanifolds of Cayley algebras, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math., 40, 17-21, (2001) · Zbl 1059.53045 [38] Banaru, MB, On the holomorphic bisectional curvature of six-dimensional Hermitian submanifolds of Cayley algebras, Bull. Transilv. Univ. Braşov., 8, 19-23, (2001) · Zbl 1084.53516 [39] Banaru, MB, Two theorems on cosymplectic hypersurfaces of six-dimensional Kählerian submanifolds of Cayley algebras, Bul. Ştiint¸. Univ. Politeh. Timiş., 46, 13-17, (2001) · Zbl 1052.53047 [40] Banaru, MB, A note on almost Hermitian manifolds with a $$J$$-invariant Ricci tensor, Izv. Akad. Nauk Resp. Moldova. Ser. Mat., 3, 88-92, (2001) · Zbl 1036.53019 [41] Banaru, MB, Some theorems on cosymplectic hypersurfaces of six-dimensional Hermitian submanifolds of Cayley algebras, Mat. Vesn. (Bull. Math. Soc. Serbia), 53, 103-110, (2001) · Zbl 1041.53034 [42] Banaru, MB, A note on six-dimensional $$G$$_{2}-submanifolds of Cayley algebras, An. Ştiint¸. Univ. Al. I. Cuza. Iaşi. Mat., 47, 389-396, (2001) · Zbl 1062.53044 [43] M. B. Banaru, “On the typical number of symmetric 6-dimensional Hermitian submanifolds of Cayley algebras,“ Proc. IX Int. Sci. Conf. “Mathematics. Computer. Education,”, Moscow (2002), pp. 118. · Zbl 1085.53042 [44] M. B. Banaru, “On $$W$$_{3}-manifolds satisfying the axiom of $$G$$-cosymplectic hypersurfaces,” Proc. XXIV Conf. Young Scientists, Moscow State Univ., Moscow (2002), pp. 15-19. · Zbl 0132.16702 [45] Banaru, MB, Two theorems on cosymplectic hypersurfaces of 6-dimensional Hermitian submanifolds of Cayley algebras, Izv. Vyssh. Uchebn. Zaved. Ser. Mat., 1, 9-12, (2002) [46] M. B. Banaru, “On the typical number of planar 6-dimensional Hermitian submanifolds of Cayley algebras,” in: Problems of Theor. Cybernetics. Proc. XIII Int. Conf. (O. B. Lupanov, ed.), 1, Moscow (2002), pp. 19. · Zbl 1085.53042 [47] Banaru, MB, Hermitian geometry of 6-dimensional submanifolds of Cayley algebras, Mat. Sb., 193, 3-16, (2002) [48] M. B. Banaru, “On the semi-Kählerian property of 6-dimensional almost Hermitian submanifolds of the octave algebra,“ Proc. Int. Sci. Conf. “Volga-2002” (Petrovskii Reaings), Kazan’ (2002), pp. 15. · Zbl 0884.53026 [49] Banaru, MB, On spectra of some tensors of six-dimensional Kählerian submanifolds of Cayley algebras, Stud. Univ. Babeş-Bolyai. Math., 47, 11-17, (2002) · Zbl 1027.53056 [50] M. B. Banaru, “On Kenmotsu hypersurfaces in a six-dimensional Hermitian submanifolds of Cayley algebras,“ in: Proc. Int. Conf. “Contemporary Geometry and Related Topics,” Beograd (2002), p. 5. · Zbl 1066.53058 [51] Banaru, MB, A note on $$R$$_{2}- and CR_{2}-manifolds, J. Harbin Inst. Tech., 9, 136-138, (2002) · Zbl 1039.53078 [52] Banaru, MB, A note on six-dimensional $$G$$1-submanifolds of the octave algebra, Taiwanese J. Math., 6, 383-388, (2002) · Zbl 1030.53061 [53] Banaru, MB, Six-dimensional Hermitian submanifolds of Cayley algebras and $$u$$-Sasakian hypersurfaces axiom, Izv. Akad. Nauk Resp. Moldova. Ser. Mat., 2, 71-76, (2002) · Zbl 1047.53042 [54] Banaru, MB, On totally umbilical cosymplectic hypersurfaces of six-dimensional Hermitian submanifolds of Cayley algebras, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math., 41, 7-12, (2002) · Zbl 1066.53058 [55] Banaru, MB, Some remarks on para-Kählerian and $$C$$-para-Kählerian manifolds, Bull. Transilv. Univ. Braşov., 9, 11-18, (2002) [56] Banaru, MB, On the type number of six-dimensional planar Hermitian submanifolds of Cayley algebras, Kyungpook Math. J., 43, 27-35, (2003) · Zbl 1050.53037 [57] Banaru, MB, A note on para-Kählerian manifolds, Kyungpook Math. J., 43, 49-61, (2003) · Zbl 1056.53044 [58] M. B. Banaru, “A note on Kirichenko tensors,” in: The Newest Problems of the Field Theory [in Russian], Kazan’ (2003), pp. 56-62. · Zbl 1017.53032 [59] Banaru, MB, On cosymplectic hypersurfaces of 6-dimensional Kählerian submanifolds of Cayley algebras, Izv. Vyssh. Uchebn. Zaved. Ser. Mat., 7, 59-63, (2003) [60] M. B. Banaru, “On the geometry of cosymplectic hypersurfaces of 6-dimensional Hermitian submanifolds of Cayley algebras,” in: The Newest Problems of the Field Theory [in Russian], Kazan’ (2003), pp. 38-43. · Zbl 0183.50803 [61] M. B. Banaru, “On the eight Gray-Hervella classes of almost Hermitian structures realized on 6-dimensional submanifolds of Cayley algebras,” The Newest Problems of the Field Theory [in Russian], Kazan’ (2003), pp. 44-50. [62] Banaru, MB, On 6-dimensional $$G$$_{2}-submanifolds of Cayley algebras, Mat. Zametki, 74, 323-328, (2003) [63] Banaru, MB, On an hypersurfaces of 6-dimensional Hermitian submanifolds of Cayley algebras, Mat. Sb., 194, 13-24, (2003) [64] Banaru, MB, On the typical number of cosymplectic hypersurfaces of 6-dimensional Hermitian submanifolds of Cayley algebras, Sib. Mat. Zh., 44, 981-991, (2003) · Zbl 1080.53048 [65] M. B. Banaru, “On Kenmotsu hypersurfaces of 6-dimensional Hermitian submanifolds of Cayley algebras,” in: Differential Geometry of Manifolds of Figures [in Russian], 34, Kaliningrad State Univ., Kaliningrad (2003), pp. 12-21. · Zbl 1086.53090 [66] M. B. Banaru, “On Kenmotsu hypersurfaces in a six-dimensional Hermitian submanifold of Cayley algebras,“ Proc. of the Workshop “Contemporary Geometry and Related Topics,” Belgrade, Yugoslavia May 15-21, 2002, World Scientific, Singapore (2004), pp. 33-40. · Zbl 1086.53080 [67] Banaru, MB, On the gray-hervella classes of AH-structures on six-dimensional submanifolds of Cayley algebras, Ann. Univ. Sofia Fac. Math. Inform., 95, 125-131, (2004) · Zbl 1080.53064 [68] M. B. Banaru, “On some almost contact metric hypersurfaces in six-dimensional special Hermitian submanifolds of Cayley algebras,“ Proc. Int. Conf. “Selected Questions of Contemporary Mathematics” Dedicated to the 200th Anniversary of C. Jacobi, Kaliningrad (2005), pp. 6. · Zbl 0351.53040 [69] M. B. Banaru, “New results of the geometry of almost Kählerian manifolds,” in: Proc. XV Military-Scientific Conf., 4, Smolensk (2007), pp. 88-90. [70] M. Banaru and G. Banaru, “On six-dimensional planar Hermitian submanifolds of Cayley algebras,” Bul. Ştiint¸. Univ. Politeh. Timiş., 46 (60), No. 1, 13-17 (2001). · Zbl 1052.53048 [71] Banaru, MB; Kirichenko, VF, Hermitian geometry of 6-dimensional submanifolds of Cayley algebras, Usp. Mat. Nauk, 49, 205-206, (1994) [72] Belgun, F; Moroianu, A, Nearly Kähler 6-manifolds with reduced holonomy, Ann. Global Anal. Geom., 19, 307-319, (2001) · Zbl 0992.53037 [73] A. L. Besse, Einstein Manifolds, Springer-Verlag, Berlin (1987). · Zbl 0613.53001 [74] Brown, R; Gray, A, Vector cross products, Comment. Math. Helv., 42, 222-236, (1967) · Zbl 0155.35702 [75] Bryant, RL, Submanifolds and special structures on the octonions, J. Differ. Geom., 17, 185-232, (1982) · Zbl 0526.53055 [76] Calabi, E, Construction and properties of some 6-dimensional almost complex manifolds, Trans. Am. Math. Soc., 87, 407-438, (1958) · Zbl 0080.37601 [77] ´E. Cartan, Le¸cons sur la g´eom´etrie des espaces de Riemann, Gauthiers-Villars (1928). · Zbl 0951.53041 [78] X. Chen, “Recent progress in Kähler geometry,” in: Proc. Int. Congr. Mat. 2002, 2, 273-282 (2002). · Zbl 1040.53083 [79] Cho, JT; Sekigawa, K, Six-dimensional quasi-Kählerian manifolds of constant sectional curvature, Tsukuba J. Math., 22, 611-627, (1998) · Zbl 0937.53033 [80] Choi, T; Lu, Z, On the DDVV conjecture and comass in calibrated geometry, I, Math. Z., 260, 409-429, (2008) · Zbl 1180.53055 [81] Cortes, V, Special Kaehler manifolds: a survey, Rend. Circ. Mat. Palermo, 69, 11-18, (2002) · Zbl 1039.53079 [82] Deszcz, R; Dillen, F; Verstraelen, L; Vrancken, L, Quasi-Einstein totally real submanifolds of nearly Kähler 6-sphere, Tôhoku Math. J., 51, 461-478, (1999) · Zbl 0990.53014 [83] Draghici, TC, On some 4-dimensional almost Kähler manifolds, Kodai Math. J., 18, 156-163, (1995) · Zbl 0836.53028 [84] Draghici, TC, Almost Kähler 4-manifolds with $$J$$-invariant Ricci tensor, Houston J. Math., 25, 133-145, (1999) · Zbl 0983.53045 [85] S. Dragomir and L. Ornea, Locally Conformal Kähler Geometry, Progr. Math., Birkhäuser, Boston (1998). · Zbl 0887.53001 [86] B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Contemporary Geometry. Methods and Applications [in Russian], Nauka, Moscow (1986). [87] B. Eckmann, “Stetige losungen linearer gleichungsysteme,” Comment. Math. Helv., 15, 318-339 (1942-1943). · Zbl 0028.32001 [88] Ejiri, N, Totally real submanifolds in a 6-sphere, Proc. Am. Math. Soc., 83, 759-763, (1981) · Zbl 0474.53051 [89] H. Freudenthal, Oktaven, Ausnahmegruppen und Oktavengeometrie, Mathematisch Instituut der Rijksuniversiteit te Utrecht (1951). · Zbl 0423.53030 [90] Funabashi, S; Pak, JS, Tubular hypersurfaces of the nearly Kähler 6-sphere, Saitama Math. J., 19, 13-36, (2001) · Zbl 1019.53024 [91] Ganchev, G; Kassabov, O, Hermitian manifolds of pointwise constant antiholomorphic sectional curvatures, Serdica Math. J., 33, 377-386, (2007) · Zbl 1199.53055 [92] G. Gheorghiev and V. Oproiu, Varietati diferentiabile finit si infinit dimensionale, Bucuresti Acad. RSR (1976-1979). · Zbl 0365.58001 [93] Goldberg, S; Kobayashi, S, Holomorphic bisectional curvature, J. Differ. Geom., 1, 225-233, (1967) · Zbl 0169.53202 [94] Gray, A, Minimal varieties and almost Hermitian submanifolds, Michigan Math. J., 12, 273-287, (1965) · Zbl 0132.16702 [95] Gray, A, Some examples of almost Hermitian manifolds, Ill. J. Math., 10, 353-366, (1966) · Zbl 0183.50803 [96] Gray, A, Six-dimensional almost complex manifolds defined by means of three-fold vector cross products, Tôhoku Math. J., 21, 614-620, (1969) · Zbl 0192.59002 [97] Gray, A, Vector cross products on manifolds, Trans. Am. Math. Soc., 141, 465-504, (1969) · Zbl 0182.24603 [98] Gray, A, Almost complex submanifolds of the six sphere, Proc. Am. Math. Soc., 20, 277-280, (1969) · Zbl 0165.55803 [99] Gray, A, Nearly Kähler manifolds, J. Differ. Geom., 4, 283-309, (1970) · Zbl 0201.54401 [100] Gray, A, The structure of nearly Kähler manifolds, Math. Ann., 223, 223-248, (1976) · Zbl 0345.53019 [101] Gray, A, Curvature identities for Hermitian and almost Hermitian manifolds, Tôhoku Math. J., 28, 601-612, (1976) · Zbl 0351.53040 [102] Gray, A; Hervella, LM, The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl., 123, 35-58, (1980) · Zbl 0444.53032 [103] Hashimoto, H, Characteristic classes of oriented 6-dimensional submanifolds in the octonions, Kodai Math. J., 16, 65-73, (1993) · Zbl 0804.57018 [104] Hashimoto, H, Oriented 6-dimensional submanifolds in the octonions, Int. J. Math. Math. Sci., 18, 111-120, (1995) · Zbl 0827.53044 [105] Hashimoto, H; Koda, T; Mashimo, K; Sekigawa, K, Extrinsic homogeneous Hermitian 6-dimensional submanifolds in the octonions, Kodai Math. J., 30, 297-321, (2007) · Zbl 1149.53034 [106] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Pure Appl. Math., 80, Academic Press, New York-San Francisco-London (1978). · Zbl 0451.53038 [107] Hernandez-Lamoneda, L, Curvature vs almost Hermitian structures, Geom. Dedic., 79, 205-218, (2000) · Zbl 0961.53041 [108] Hervella, LM; Vidal, E, Novelles gèomètries pseudo-kahlèriennes $$G$$_{1} et $$G$$_{2}, C. R. Acad. Sci. Paris. Ser., 1, 115-118, (1976) [109] C. C. Hsiung, Almost Complex and Complex Structures, World Scientific, Singapore (1995). · Zbl 0838.53050 [110] N. E. Hurt, Geometric Quantization in Action. Applications of Harmonic Analysis in Quantum Statistical Mechanics and Quantum Field Theory, Math. Appl., 8, Reidel Publ., Dordrecht-Boston-London (1983). · Zbl 0505.22007 [111] Ianus, S, Submanifolds of almost Hermitian manifolds, Riv. Mat. Univ. Parma, 3, 123-142, (1994) · Zbl 0846.53010 [112] S. Ianus, Geometrie Diferentiala cu Aplicatii in Teoria Relativitatii, Editura Academiei Romane, Bucure,sti (1983). · Zbl 0542.53001 [113] J. Jost, Riemannian Geometry and Geometric Analysis, Springer-Verlag, Berlin-Heidelberg-New York (2003). · Zbl 0828.53002 [114] Kashiwada, T, On a class of locally conformal Kähler manifolds, Tensor (N.S.), 63, 297-306, (2002) · Zbl 1119.53046 [115] Kim, HS; Takagi, R, The type number of real hypersurfaces in $$P$$_{$$n$$}($$C$$), Tsukuba J. Math., 20, 349-356, (1996) · Zbl 0906.53036 [116] Un Kyu Kim, “On six-dimensional almost Hermitian manifolds with pointwise constant holomorphic sectional curvature,” Nihonkai Math. J., 6, 185-200 (1995). · Zbl 0997.53501 [117] Kirichenko, VF, Almost Kählerian structures induced by 3-vector products on 6-dimensional submanifolds of Cayley algebras, Vestn. MGU. Ser. Mat. Mekh., 3, 70-75, (1973) [118] Kirichenko, VF, $$K$$-spaces of the constant type, Sib. Mat. Zh., 17, 282-289, (1976) · Zbl 0329.53021 [119] Kirichenko, VF, Classification of Kählerian structures induced by 3-vector products on 6-dimensional submanifolds of Cayley algebras, Izv. Vyssh. Uchebn. Zaved. Ser. Mat., 8, 32-38, (1980) · Zbl 0449.53049 [120] Kirichenko, VF, Stability of almost Hermitian structures induced by 3-vector products on 6-dimensional submanifolds of Cayley algebras, Ukr. Geom. Sb., 25, 60-68, (1982) · Zbl 0508.53045 [121] Kirichenko, VF, The tangent bundle from the point of view of generalized Hermitian geometry, Izv. Vyssh. Uchebn. Zaved. Ser. Mat., 6, 50-58, (1984) · Zbl 0553.53021 [122] V. F. Kirichenko, “Methods of generalized Hermitian geometry in the theory of almost contact manifolds,” in: Itogi Nauki Tekhn. Probl. Geom., 18, All-Russian Institute for Scientific and Technical Information (VINITI), Moscow (1986), pp. 25-71. [123] Kirichenko, VF, Locally conformal Kählerian manifolds of a constant holomorphic sectional curvature, Mat. Sb., 182, 354-362, (1991) · Zbl 0732.53055 [124] Kirichenko, VF, Hermitian geometry of 6-dimensional symmetric submanifolds of Cayley algebras, Vestn. MGU. Ser. Mat. Mekh., 1, 6-13, (1994) [125] V. F. Kirichenko, Differential-Geometric Structures on Manifolds [in Russian], Moscow (2003). · Zbl 0990.53014 [126] Kirichenko, VF, Generalized gray-hervella classes and holomorphically-projective trnsformations of almost Hermitian structures, Izv. Ross. Akad. Nauk. Ser. Mat., 69, 107-132, (2005) [127] Kirichenko, VF; Ezhova, NA, Conformal invariants of Vaisman-gray manifolds, Usp. Mat. Nauk, 51, 163-164, (1996) · Zbl 0884.53025 [128] Kirichenko, VF; Shchipkova, NN, On the geometry of Vaisman-gray manifolds, Usp. Mat. Nauk, 49, 155-156, (1994) [129] Kirichenko, VF; Vlasova, LI, Concircular geometry of approximately Kählerian manifolds, Mat. Sb., 193, 51-76, (2002) [130] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. 1, Interscience Publ. New York-London (1963). · Zbl 0119.37502 [131] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. 2, Interscience Publ. New York-London (1969). · Zbl 0175.48504 [132] M. Kon and K. Yano, Structures on Manifolds, Pure Math., 3, World Scientific (1984). · Zbl 0983.53045 [133] Kurihara, H, On real hypersurfaces in a complex space form, Math. J. Okayama Univ., 40, 177-186, (1998) · Zbl 0951.53041 [134] Kurihara, H, The type number on real hypersurfaces in a quaternionic space form, Tsukuba J. Math., 24, 127-132, (2000) · Zbl 1034.53061 [135] Kurihara, H; Takagi, R, A note on the type number of real hypersurfaces in $$P$$_{$$n$$}($$C$$), Tsukuba J. Math., 22, 793-802, (1998) [136] G. F. Laptev, “Fundamental higher-orders infinitesimal structures on smooth manifolds,” in: Tr. Geom. Semin., 1, All-Union Institute for Scientific and Technical Information (VINITI), Moscow (1966), pp. 139-189. · Zbl 0171.42301 [137] Levko, JJ, Some characterizations of Kählerian structure, Tensor (N.S.), 41, 249-257, (1984) · Zbl 0582.53045 [138] Levko, JJ, Almost semi-Kählerian structure, Tensor (N.S.), 64, 295-296, (2003) · Zbl 1165.53339 [139] Li, H, The Ricci curvature of totally real 3-dimensional submanifolds of the nearly Kaehler 6-sphere, Bull. Belg. Math. Soc. Simon Stevin., 3, 193-199, (1996) · Zbl 0853.53037 [140] Li, H; Wei, G, Classification of Lagrangian Willmore submanifolds of the nearly Kaehler 6-sphere $$S$$_{1}\^{6} with constant scalar curvature, Glasgow Math. J., 48, 53-64, (2006) · Zbl 1117.53021 [141] A. Lichnerowicz, Th´orie globale des connexions et des groupes d’holonomie, Rome, Edizioni Cremonese (1955). · Zbl 0116.39101 [142] Luczyszyn, D, On para-Kählerian manifolds with recurrent paraholomorphic projective curvature, Math. Balkanica (N.S.), 14, 167-176, (2000) · Zbl 1330.53039 [143] Luczyszyn, D, On Bochner semisymmetric para-Kählerian manifolds, Demonstr. Math., 4, 933-942, (2001) · Zbl 1029.53038 [144] Luczyszyn, D, On pseudosymmetric para-Kählerian manifolds, Beiträge Algebra Geom., 44, 551-558, (2003) · Zbl 1076.53034 [145] Mari, G, Curvature identities for an almost $$C$$-manifold, Stud. Cerc. Mat., 50, 23-38, (1998) · Zbl 1017.53032 [146] Matsumoto, M, On 6-dimensional almost tachibana spaces, Tensor (N.S.), 23, 250-252, (1972) · Zbl 0236.53048 [147] Matzeu, P; Munteanu, M-I, Vector cross products and almost contact structures, Rend. Mat. Roma, 22, 359-376, (2002) · Zbl 1051.53023 [148] A. S. Mishchenko and A. T. Fomenko, A Course in Differential Geometry and Topology [in Russian], Moscow (1980). · Zbl 0524.53001 [149] Mishra, RS, Normality of the hypersurfaces of almost Hermite manifolds, J. Indian Math. Soc., 61, 71-79, (1995) · Zbl 0857.53014 [150] Munteanu, MI, Doubly warped products CR-submanifolds in locally conformal Kaehler manifolds, Monatsh. Math., 150, 333-342, (2007) · Zbl 1128.53030 [151] Nagy, P-A, On nearly-Kähler geometry, Ann. Global Anal. Geom., 22, 167-178, (2002) · Zbl 1020.53030 [152] Nakagawa, H; Takagi, R, On locally symmetric Kaehler submanifolds in a complex projective space, J. Math. Soc. Jpn., 28, 638-667, (1976) · Zbl 0328.53009 [153] Nannicini, A, On certain Kähler submanifolds of twistor spaces, Boll. Unione Mat. Ital. Sez. B, 11, 257-265, (1997) · Zbl 0884.53026 [154] A. P. Norden, Theory of Surfaces [in Russian], Moscow (1956). [155] Oproiu, V, Some classes of natural almost Hermitian structures on the tangent bundles, Publ. Math. Debrecen., 62, 561-576, (2003) · Zbl 1026.53038 [156] Oproiu, V, Some classes of general natural almost Hermitian structures on tangent bundles, Rev. Roum. Math. Pures Appl., 48, 521-533, (2003) · Zbl 1099.53049 [157] Panak, M; Vanzura, J, Three-forms and almost complex structures on six-dimensional manifolds, J. Austr. Math. Soc., 84, 247-263, (2008) · Zbl 1156.53017 [158] V. I. Pan’zhenskii and K. B. Shiryaev, “Tensor signs of classes of almost Hermitian structures on the tangent bundle,” in: Motions in Generalized Spaces [in Russian], Penza (1999), pp. 126-132. [159] A. Z. Petrov, Einstein Spaces [in Russian], Moscow (1961). [160] M. M. Postnikov, Lectures in Geometry. Semester IV. Differential Geometry [in Russian], Nauka, Moscow (1988). · Zbl 0659.53001 [161] P. K. Rashevskii, Riemannian Geometry and Tensor Analysis [in Russian], Nauka, Moscow (1967). · Zbl 0186.06502 [162] Rizza, GB, Varieta parakähleriane, Ann. Mat. Pura Appl., 98, 47-61, (1974) · Zbl 0279.53051 [163] Ryan, PJ, Kähler manifolds as real hypersurfaces, Duke Math. J., 40, 207-213, (1973) · Zbl 0257.53055 [164] Sato, T, An example of an almost Kähler manifold with pointwise constant holomorphic sectional curvature, Tokyo J. Math., 23, 387-401, (2000) · Zbl 0981.53026 [165] Sawaki, S; Sekigawa, K, Almost Hermitian manifolds with constant holomorphic sectional curvature, J. Differ. Geom., 9, 123-134, (1974) · Zbl 0277.53036 [166] Sekigawa, K, Almost Hermitian manifolds satisfying some curvature conditions, Kodai Math. J., 2, 384-405, (1979) · Zbl 0423.53030 [167] Sekigawa, K, Almost complex submanifolds of a six-dimensional sphere, Kodai Math. J., 6, 174-185, (1983) · Zbl 0517.53055 [168] Sekigawa, K, On some compact Einstein almost Kähler manifolds, J. Math. Soc. Jpn., 36, 677-684, (1987) · Zbl 0637.53053 [169] Sekigawa, K, On some 4-dimensional compact almost Hermitian manifolds, J. Ramanujan Math. Soc., 2, 101-116, (1987) · Zbl 0668.53019 [170] S. S. Shern, M. P. Do Carmo, and S. Kobayashi, “Minimal submanifolds of a sphere with second fundamental form of constant length,” in: Functional Analysis and Related Fields, Springer- Verlag, Berlin (1970), pp. 59-75. · Zbl 0257.53055 [171] Takagi, R, A class of hypersurfaces with constant principal curvatures in a sphere, J. Differ. Geom., 11, 225-233, (1976) · Zbl 0337.53003 [172] Tang, Z, Curvature and integrability of an almost Hermitian structure, Int. J. Math., 27, 97-105, (2006) · Zbl 1111.53027 [173] Tanno, S, Constancy of holomorphic sectional curvature in almost Hermitian manifolds, Kodai Math. Semin. Repts., 25, 190-201, (1973) · Zbl 0263.53019 [174] Tanno, S, Ricci curvature of contact Riemannian manifolds, Tôhoku Math. J., 40, 441-448, (1988) · Zbl 0655.53035 [175] Tekkoyun, M, A general view to classification of almost Hermitian manifolds, Rend. Inst. Mat. Univ. Trieste, 38, 1-15, (2006) · Zbl 1138.32013 [176] Tricerri, F, Some examples of locally conformal Kähler manifolds, Rend. Sem. Mat. Univ. Politec. Torino, 40, 81-92, (1982) · Zbl 0511.53068 [177] Tricerri, F; Vanhecke, L, Curvature tensors on almost Hermitian manifolds, Trans. Am. Math. Soc., 267, 365-398, (1981) · Zbl 0484.53014 [178] Vaisman, I, On locally conformal almost Kähler manifolds, Israel J. Math., 24, 338-351, (1976) · Zbl 0335.53055 [179] Vaisman, I, On locally and globally conformal Kähler manifolds, Trans. Am. Math. Soc., 2, 533-542, (1980) · Zbl 0446.53048 [180] L. Vanhecke, “Almost Hermitian manifolds with $$J$$-invariant Riemann curvature tensor,” Rend. Sem. Mat. Univ. Politec. Torino, 34, 487-498 (1975-1977). · Zbl 0511.53068 [181] Vanhecke, L, The Bochner curvature tensor on almost Hermitian manifolds, Geom. Dedic., 6, 389-397, (1977) · Zbl 0383.53022 [182] Vrancken, L, Special Lagrangian submanifolds of the nearly Kaehler 6-sphere, Glasgow Math. J., 45, 415-426, (2003) · Zbl 1053.53050 [183] Watson, B, New examples of strictly almost Kähler manifolds, Proc. Am. Math. Soc., 88, 541-544, (1983) · Zbl 0517.53039 [184] Whitehead, G, Note on cross-sections in Stiefel manifolds, Comment. Math. Helv., 34, 239-240, (1962) · Zbl 0118.18702 [185] K. Yano, Differential Geometry on Complex and Almost Complex Spaces, Pergamon Press, Oxford (1965). · Zbl 0127.12405 [186] Yano, K; Ishihara, S, Almost contact structures induced on hypersurfaces in complex and almost complex spaces, Kodai Math. Sem. Rep., 17, 222-249, (1965) · Zbl 0132.16802 [187] K. Yano and M. Kon, Structures on Manifolds, World Scientific, Singapore (1984). · Zbl 0557.53001 [188] K. Yano and T. Sumitomo, “Differential geometry of hypersurfaces in a Cayley space,” Proc. Roy. Soc. Edinburgh. Sec. A, 66, 216-231 (1962-1964). · Zbl 0188.54404
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.