# zbMATH — the first resource for mathematics

Almost contact metric structures on the hypersurface of almost Hermitian manifolds. (English. Russian original) Zbl 1325.53037
J. Math. Sci., New York 207, No. 4, 513-537 (2015); translation from Itogi Nauki Tekh., Ser. Sovrem. Mat. Prilozh., Temat. Obz. 127 (2014).
This is a review paper on the theories of almost contact, almost Hermitian and Kenmotsu structures and their use in classical differential geometry. The authors give a very comprehensive account on the historical development of these structures using only Riemannian metrics, although modern applicable studies on these structures have been extended using semi-Riemannian and degenerate metrics for the purpose of their applications. Also, there is a long list of 200 references (which include some minor side references), but, they miss to include the fundamental works of Chen, a 1989 paper of Grey on some global properties of contact structures and S. MacLane [Reports of the Midwest Category Seminar. II. York: Springer-Verlag (1968; Zbl 0165.00201)] on physical applications of contact structures.

##### MSC:
 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 53C20 Global Riemannian geometry, including pinching 53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
Full Text:
##### References:
 [1] Abu-Saleem, A; Banaru, M, Some applications of kirichenko tensors, An. Univ. Oradea, Fasc. Mat., 17, 201-208, (2010) · Zbl 1212.53045 [2] Adachi, T; Kameda, M; Maeda, S, Real hypersurfaces which are contact in a nonflat complex space form, Hokkaido Math. J., 40, 205-217, (2011) · Zbl 1232.53045 [3] Alegre, P, Semi-invariant submanifolds of lorentzian Sasakian manifolds, Demonstr. Math., 44, 391-406, (2011) · Zbl 1250.53049 [4] Apostolov, V; Ganchev, G; Inanov, S, Compact Hermitian surfaces of constant antiholomorphic curvature, Proc. Am. Math. Soc., 125, 3705-3714, (1997) · Zbl 0898.53025 [5] V. I. Arnold, Mathematical Methods of Classical Mechanics [in Russian], Nauka, Moscow (1989). [6] Arslan, K; De, UC; Ozgur, C; Jun, J-B, On a class of Sasakian manifolds, Nihonkai Math. J., 19, 21-27, (2008) · Zbl 1165.53028 [7] Bagewadi, CS; Girish Kumar, E, Note on trans-Sasakian manifolds, Tensor (N.S.), 65, 80-88, (2004) · Zbl 1165.53343 [8] Bagewadi, CS; Prasada, VS, Note on Kenmotsu manifolds, Bull. Calcutta Math. Soc., 91, 379-384, (1999) · Zbl 0977.53512 [9] 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. · Zbl 0116.38901 [10] M. B. Banaru, “Gray-Hervella classes of almost Hermitian structures on 6-dimensional submanifolds of the Cayley algebra,” in: Nauch. Tr. MGPU im. V. I. Lenina, Prometey, Moscow (1994), pp. 36-38. · Zbl 0112.14002 [11] M. B. Banaru “On spectra of the most important tensors of 6-dimensional Hermitian submanifolds of the Cayley algebra,” in: The Newest Problems of the Field Theory [in Russian], Kazan’ (2000), pp. 18-22. · Zbl 1044.53020 [12] M. B. Banaru, “The axiom of $$g$$-cosymplectic hypersurfaces for 6-dimensional Hermitian submanifolds of the octave algebra,” in: Boundary-Value Problems in Complex Analysis and Differential Equations, Vol. 2, Smolensk (2000), pp. 36-41. · Zbl 1224.53063 [13] Banaru, M, Six theorems on six-dimensional Hermitian submanifolds of Cayley algebra, Izv. Akad. Nauk Resp. Moldova, Ser. Mat., 3, 3-10, (2000) · Zbl 1031.53087 [14] 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 [15] M. B. Banaru, “On the geometry of cosymplectic hypersurfaces of 6-dimensional submanifolds of the Cayley algebra,“ in: Proc. Int. Conf. “Volga-2001” (Perovskii Readings), Kazan’ (2001), p. 25. · Zbl 1079.53506 [16] Banaru, MB, Two theorems on cosymplectic hypersurfaces of six-dimensional Hermitian submanifolds of Cayley algebra, J. Harbin Inst. Tech., 8, 38-40, (2001) [17] 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 Applications, Opava, Czech Republic (2001), p. 4. · Zbl 0166.17801 [18] 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. · Zbl 0908.53024 [19] Banaru, MB, Two theorems on cosymplectic hypersurfaces of six-dimensional Kählerian submanifolds of Cayley algebras, Bul. Ştiinţ. Univ. Politeh. Timiş., 46, 13-17, (2001) · Zbl 1052.53047 [20] 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 [21] 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 0853.53037 [22] 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) [23] M. B. Banaru, “On the typical number of cosymplectic hypersurfaces of a 6-dimensional Kählerian submanifold of the Cayley algebra,“ in: Proc. X Int. Conf. “Mathematics. Economics. Education,” and II Int. Symp. “Fourier Series and Their Applications,” Rostov-on-Don (2002), pp. 116-117. · Zbl 1222.53069 [24] 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 [25] M. B. Banaru, “On Kenmotsu hypersurfaces in six-dimensional Hermitian submanifolds of Cayley algebras,“ in: Proc. Int. Conf. “Contemporary Geometry and Related Topics,” Beograd (2002), p. 5. · Zbl 1066.53058 [26] Banaru, M, On nearly-cosymplectic hypersurfaces in nearly-Kählerian manifolds, Stud. Univ. Babeş-Bolyai Math., 47, 3-11, (2002) · Zbl 1027.53082 [27] 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 [28] 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 [29] Banaru, M, On minimality of a Sasakian hypersurfaces in a $$W$$_{3}-manifold, Saitama Math. J., 20, 1-7, (2002) · Zbl 1052.53045 [30] Banaru, MB, Hermitian geometry of 6-dimensional submanifolds of Cayley algebras, Mat. Sb., 193, 3-16, (2002) [31] M. B. Banaru, “On the similarity and dissimilarity of the properties of Kenmotsu and Sasaki hypersurfaces of special Hermitian manifolds,” in: Proc. X Military-Scientific Conf., Smolensk (2002), p. 196. [32] M. B. Banaru, “On $$W$$_{4}-manifolds satisfying the axiom of $$G$$-cosymplectic hypersurfaces,“ in: Proc. Int. Sci. Conf. “Volga-2001” (Perovskii Readings), Kazan’ (2002), p. 16. · Zbl 1079.53506 [33] Banaru, MB, On the typical number of weakly cosymplectic hypersurfaces of approximately Kählerian manifolds, Fundam. Prikl. Mat., 8, 357-364, (2002) · Zbl 1036.53013 [34] Banaru, MB, On Hermitian manifolds satisfying the axiom of $$U$$-cosymplectic hypersurfaces, Fundam. Prikl. Mat., 8, 943-947, (2002) · Zbl 1055.53017 [35] M. B. Banaru, “On the geometry of weakly cosymplectic hypersurfaces of NK-manifolds,” in: Proc. Int. Semin. on Geometry and Analysis to the Memory of Prof. N. V. Efimov, Rostov-on-Don (2002), pp. 18-19. · Zbl 0449.53049 [36] Banaru, MB, On cosymplectic hypersurfaces of 6-dimensional Kählerian submanifolds of the Cayley algebra, Izv. Vyssh. Ucheb. Zaved. Ser. Mat., 7, 59-63, (2003) [37] M. B. Banaru, “On the geometry of cosymplectic hypersurfaces of 6-dimensional Hermitian submanifolds of the Cayley algebra,” in: The Newest Problems of the Field Theory [in Russian], Kazan’ (2003), pp. 38-43. [38] M. B. Banaru, “On eight Gray-Hervella classes of almost Hermitian structures realized on 6-dimensional submanifolds of the Cayley algebra,” in: The Newest Problems of the Field Theory [in Russian], Kazan’ (2003), pp. 44-50. [39] M. B. Banaru, “On $$W$$_{4}-manifolds satisfying the axiom of $$G$$-cosymplectic hypersurfaces,” in: The Newest Problems of the Field Theory [in Russian], Kazan’ (2003), pp. 51-55. · Zbl 0906.53036 [40] Banaru, M, Hermitian manifolds and $$U$$-cosymplectic hypersurfaces axiom, J. Sichuan Normal Univ. Nat. Sci. Ed., 26, 261-263, (2003) · Zbl 1068.53040 [41] M. B. Banaru, “A note on Kirichenko tensors,” in: The Newest Problems of the Field Theory [in Russian], Kazan’ (2003), pp. 56-62. · Zbl 0993.58013 [42] Banaru, MB, On Sasakian hypersurfaces of 6-dimensional Hermitian submanifolds of the Cayley algebra, Mat. Sb., 194, 13-24, (2003) [43] Banaru, MB, On the typical number of cosymplectic hypersurfaces of 6-dimensional Hermitian submanifolds of the Cayley algebra, Sib. Mat. Zh., 44, 981-991, (2003) · Zbl 1080.53048 [44] M. B. Banaru, “On almost contact metric structures on hypersurfaces of 6-dimensional Kählerian submanifolds of the octave algebra,” in: Actual Problems of Mathematical Physics [in Russian], Prometey, Moscow (2003), pp. 40-41. · Zbl 0472.53043 [45] 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 [46] Banaru, MB, On Kenmotsu hypersurfaces of special Hermitian manifolds, Sib. Mat. Zh., 45, 11-15, (2004) · Zbl 1125.53038 [47] M. B. Banaru, “On typical numbers of almost contact metric hypersurfaces of almost Hermitian submanifolds,“ in: Proc. VIII Int. Semin. “Discrete Mathematics and Its Applications,” Moscow State Univ., Moscow (2004), pp. 379-381. · Zbl 1194.53048 [48] M. B. Banaru, “On Sasakian hypersurfaces of special Hermitian manifolds,” in: Proc. XXV Conf. Young Scientists, Moscow State Univ., Moscow (2004), pp. 11-14. · Zbl 1125.53038 [49] M. Banaru, “On cosymplectic hypersurfaces in a Kählerian manifold,” in: Proc. Int. Conf. on Mathematics and Its Applications, Kuwait (2004), pp. 62-63. · Zbl 0185.25104 [50] M. 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 [51] 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 [52] M. B. Banaru, “On the axiom of Kenmotsu $$u$$-hypersurfaces for special Hermitian manifolds,” in: Proc. XIII Military-Sci. Conf., Smolensk (2005), pp. 224-225. [53] 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 1036.53029 [54] M. B. Banaru, “On almost contact metric structures on hypersurfaces of almost Hermitian manifolds of the classes $$W$$_{3} and $$W$$_{4},“ in: Proc. Rep. Int. Conf. “A. Z. Petrov Anniversary Symposium on General Relativity and Gravitation,” Kazan’ , November 1-6, 2010, Kazan’ (2010), pp. 41-42. · Zbl 0961.53041 [55] Banaru, MB; Kirichenko, VF, Hermitian geometry of 6-dimensional submanifolds of the Cayley algebra, Usp. Mat. Nauk, 1, 205-206, (1994) [56] Bejancu, A; Farran, HR, On totally umbilical QR-submanifolds of quaternion Kählerian manifolds, Bull. Austr. Math. Soc., 62, 95-103, (2000) · Zbl 0974.53015 [57] Belgun, F; Moroianu, A; Semmelmann, U, Symmetries of contact metric manifolds, Ceom. Dedic., 101, 203-216, (2003) · Zbl 1049.53032 [58] A. Besse, Einstein Manifolds, Springer-Verlag, Berlin etc. (1987). · Zbl 0613.53001 [59] Blaga, AM, Affine connections on almost para-cosymplectic manifolds, Czechoslovak Math. J., 61, 863-871, (2011) · Zbl 1249.53038 [60] Blair, DE, The theory of quasi-Sasakian structures, J. Differ. Geom., 1, 331-345, (1967) · Zbl 0163.43903 [61] D. E. Blair, Contact Manifolds in Riemannian Geometry, Lect. Notes Math., 509 (1976). · Zbl 0319.53026 [62] Blair, DE, Two remarks on contact metric structure, Tôhoku Math. J., 29, 319-324, (1977) · Zbl 0376.53021 [63] Blair, DE, A hyperbolic twistor space, Balkan J. Geom. Appl., 5, 9-16, (2000) · Zbl 0997.53024 [64] D. E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds, Progr. Math., Birkhäuser, Boston-Basel-Berlin (2002). · Zbl 1011.53001 [65] Blair, DE, A product twistor space, Serdica Math. J., 28, 163-174, (2002) · Zbl 1041.53022 [66] Blair, DE; Kim, JS; Tripathi, MM, On the concircular curvature tensor of a contact metric manifold, J. Korean Math. Soc., 42, 883-892, (2005) · Zbl 1084.53039 [67] Blair, DE; Showers, DK; Yano, K, Nearly Sasakian structures, Kodai Math. Semin. Repts., 27, 175-180, (1976) · Zbl 0328.53036 [68] Borthwick, D; Uribe, A, Almost complex structures and geometric quantization, Math. Res. Lett., 3, 845-861, (1996) · Zbl 0872.58030 [69] Bryant, RL, Submanifolds and special structures on the octonions, J. Differ. Geom., 17, 185-232, (1982) · Zbl 0526.53055 [70] Calin, C, On the geometry of hypersurfaces in a quasi-Sasakian manifold, Demonstr. Math., 36, 451-462, (2003) · Zbl 1038.53053 [71] Calin, C, Kenmotsu manifolds with $$η$$-parallel Ricci tensor, Bull. Soc. Math. Banja Luka, 10, 10-15, (2003) · Zbl 1060.53052 [72] C. Calin, Subvarietati in varietati aproape de contact, Performantica, Iasi (2005). · Zbl 1093.53056 [73] Cho, JT, A new class of contact Riemannian manifolds, Isr. J. Math., 109, 299-318, (1999) · Zbl 0944.53023 [74] Cho, JT, Ricci solitons and odd-dimensional spheres, Monatsh. Math., 160, 347-357, (2010) · Zbl 1219.53037 [75] Choi, T; Lu, Z, On the DDVV conjecture and comass in calibrated geometry, I, Math. Z., 260, 409-429, (2008) · Zbl 1180.53055 [76] E. Cartan, Riemannian Geometry in an Orthogonal Frame [Russian translation], Moscow State Univ., Moscow (1960). [77] De, UC; Pathak, G, Torseforming vector fields in a Kenmotsu manifold, An. Ştinţ. Univ. Al. I. Cuza, Iaşi, 49, 257-264, (2003) · Zbl 1066.53128 [78] De, UC; Pathak, G, On 3-dimensional Kenmotsu manifolds, Indian J. Pure Appl. Math., 35, 159-165, (2004) · Zbl 1061.53011 [79] De, UC; Avijit Sarkar, On $$ϕ$$-Ricci symmetric Sasakian manifolds, Proc. Jangjeon Math. Soc., 11, 47-52, (2008) · Zbl 1146.53028 [80] De, UC; Shaikh, AA; Biswas Sudipta, On φ-recurrent Sasakian manifolds, Novi Sad J. Math., 33, 43-48, (2003) · Zbl 1090.53033 [81] De, UC; Shaikh, AA; Biswas Sudipta, On weakly symmetric contact metric manifolds, Tensor (N.S.), 64, 170-175, (2003) · Zbl 1165.53370 [82] Deszcz, R; Hotlos, M, On hypersurfaces with type number two in spaces of constant curvature, Ann. Univ. Sci. Budapest. Eötvös Sect. Math., 46, 19-34, (2003) · Zbl 1083.53023 [83] Dileo, G, A classification of certain almost $$α$$-Kenmotsu manifolds, Kodai Math. J., 44, 426-445, (2011) · Zbl 1241.53025 [84] Dileo, G; Pastore, AM, Almost Kenmotsu manifolds and local symmetry, Bull. Belg. Math. Soc. Simon Stevin., 14, 343-354, (2007) · Zbl 1148.53034 [85] Dileo, G; Pastore, AM, Almost Kenmotsu manifolds and nullity distributions, J. Geom., 93, 46-61, (2009) · Zbl 1204.53025 [86] Dileo, G; Pastore, AM, Almost Kenmotsu manifolds with a condition of $$η$$-parallelism, Differ. Geom. Appl., 27, 671-679, (2009) · Zbl 1183.53024 [87] B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Contemporary Geometry. Methods and Applications [in Russian], Nauka, Moscow (1986). [88] Ejiri, N, Totally real submanifolds in a 6-sphere, Proc. Am. Math. Soc., 83, 759-763, (1981) · Zbl 0474.53051 [89] Endo, H, On the curvature tensor of nearly cosymplectic manifolds of constant φ-sectional curvature, An. Ştinţ. Univ. Al. I. Cuza, Iaşi, 51, 439-454, (2005) · Zbl 1112.53061 [90] Endo, H, Remarks on nearly cosymplectic manifolds of constant φ-sectional curvature with a submersion of geodesic fibres, Tensor (N.S.), 66, 26-39, (2005) · Zbl 1127.53025 [91] Endo, H, On nearly cosymplectic manifolds of constant φ-sectional curvature, Tensor (N.S.), 67, 323-335, (2006) · Zbl 1136.53328 [92] Endo, H, Some remarks of nearly cosymplectic manifolds of constant φ-sectional curvature, Tensor (N.S.), 68, 204-221, (2007) · Zbl 1193.53176 [93] H. Freudenthal, Oktaven, Ausnahmegruppen und Oktavengeometrie, Mathematisch Instituut der Rijksuniversiteit te Utrecht (1951). · Zbl 0054.01701 [94] Fueki, S; Endo, H, On conformally flat nearly cosymplectic manifolds, Tensor (N.S.), 66, 305-316, (2005) · Zbl 1127.53038 [95] Funabashi, S; Pak, JS, Tubular hypersurfaces of the nearly Kähler 6-sphere, Saitama Math. J., 19, 13-36, (2001) · Zbl 1019.53024 [96] G. Gheorghiev and V. Oproiu, Varietati diferentiabile finit si infinit dimensionale, Bucuresti Acad. RSR (1976-1979). · Zbl 0365.58001 [97] Gherghe, C, Harmonic maps on Kenmotsu manifolds, Rev. Roumaine Math. Pures Appl., 45, 447-453, (2000) · Zbl 0993.58013 [98] Ghosh, A, Kenmotsu 3-metric as a Ricci soliton, Chaos Solitons Fractals, 44, 647-650, (2011) · Zbl 1273.37040 [99] Goldberg, S, Totally geodesic hypersurfaces of Kähler manifolds, Pac. J. Math., 27, 275-281, (1968) · Zbl 0165.24802 [100] Goldberg, S; Yano, K, Integrability of almost cosymplectic structures, Pac. J. Math., 31, 373-382, (1969) · Zbl 0185.25104 [101] Gouli-Andreou, F; Tsolakidou, N, On conformally flat contact metric manifolds, J. Geom., 79, 75-88, (2004) · Zbl 1064.53028 [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] Gupta, RS; Haider, SMK; Shahid, MN, Slant submanifolds of cosymplectic manifolds, An. Ştinţ. Univ. Al. I. Cuza, Iaşi, 50, 33-50, (2004) · Zbl 1074.53069 [104] Gupta, RS; Sharfuddin, A, Screen transversal lightlike submanifolds of indefinite Kenmotsu manifolds, Sarajevo J. Math., 7, 103-113, (2011) · Zbl 1222.53018 [105] Hamada, T; Inoguchi, J, Ruled real hypersurfaces of complex space forms, Kodai Math. J., 33, 123-134, (2010) · Zbl 1194.53048 [106] Hatakeyama, Y; Ogawa, Y; Tanno, S, Some properties of manifolds with contact metric structures, Tˆohoku Math. J., 15, 42-48, (1963) · Zbl 0196.54902 [107] Hernandez-Lamoneda, L, Curvature vs. almost Hermitian structure, Geom. Dedic., 79, 205-218, (2000) · Zbl 0961.53041 [108] Hristov, M, On the locally quasi-Sasakian manifolds, C. R. Acad. Bulgare Sci., 56, 9-14, (2003) · Zbl 1036.53029 [109] 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 [110] Ianus, S, Submanifolds of almost Hermitian manifolds, Riv. Mat. Univ. Parma, 3, 123-142, (1994) · Zbl 0846.53010 [111] S. Ianus, Geometrie Diferentiala cu Aplicatii in Teoria Relativitatii, Editura Academiei Romane, Bucureşti (1983). · Zbl 0542.53001 [112] Jamal, N; Khan, KA; Khan, VA, Generic warped product submanifolds of locally conformal Kähler manifolds, Acta Math. Sci., 30B, 1457-1468, (2010) · Zbl 1240.53088 [113] Janssens, D; Vanhecke, L, Almost contact structures and curvature tensors, Kodai Math. J., 4, 1-27, (1981) · Zbl 0472.53043 [114] J. Jost, Riemannian Geometry and Geometric Analysis, Springer-Verlag, Berlin-Heidelberg-New York (2003). · Zbl 0828.53002 [115] Kenmotsu, K, A class of almost contact Riemannian manifolds, Tôhoku Math. J., 24, 93-103, (1972) · Zbl 0245.53040 [116] Q. Khan, “On an Einstein projective Sasakian manifold,” İstanbul Üniv. Fen Fak. Mat. Derg., 61-62, 97-103 (2002-2003). [117] Kim, HS; Kim, I-B; Takagi, R, Extrinsically homogeneous real hypersurfaces with three distinct principal curvatures in $$H$$_{$$n$$}($$C$$), Osaka J. Math., 41, 853-863, (2004) · Zbl 1094.53051 [118] Kim, HS; Takagi, R, The type number of real hypersurfaces in $$P$$_{$$n$$}($$C$$), Tsukuba J. Math., 20, 349-356, (1996) · Zbl 0906.53036 [119] Kirichenko, VF, Almost Kählerian structures induced by 3-vector products on 6-dimensional submanifolds of the Cayley algebra, Vestn. MGU, Ser. Mat. Mekh., 3, 70-75, (1973) [120] Kirichenko, VF, A classification of Kählerian structures induced by 3-vector products on 6-dimensional submanifolds of the Cayley algebra, Izv. Vyssh. Ucheb. Zaved. Ser. Mat., 8, 32-38, (1980) · Zbl 0449.53049 [121] Kirichenko, VF, The stability of almost Hermitian structures induced by 3-vector products on 6-dimensional submanifolds of the Cayley algebra, Ukr. Geom. Sb., 25, 60-68, (1982) · Zbl 0508.53045 [122] V. F. Kirichenko, “The methods of the generalized Hermitian geometry in the theory of almost contact manifolds,” in: Itogi Nauki Tekhn. Probl. Geom., 18, All-Union Institute for Scientific and Technical Information (VINITI), Moscow (1986), pp. 25-71. · Zbl 0274.53062 [123] Kirichenko, VF, Hermitian geometry of 6-dimensional symmetric submanifolds of the Cayley algebra, Vestn. MGU, Ser. Mat. Mekh., 3, 6-13, (1994) [124] Kirichenko, VF, Differential geometry of principal toroidal bundles, Fundam. Prikl. Mat., 6, 1095-1120, (2000) · Zbl 1124.53303 [125] V. F. Kirichenko, “On the geometry of Kenmotsu manifolds,” Dokl. Ross. Akad. Nauk, 380, No. 5, 585-587. · Zbl 1044.53055 [126] V. F. Kirichenko, Differential-Geometric Structures on Manifolds, [in Russian], Moscow (2003). · Zbl 1250.53049 [127] Kirichenko, VF; Rodina, EV, On the geometry of trans-Sasakian and almost trans-Sasakian manifolds, Fundam. Prikl. Mat., 3, 837-846, (1997) · Zbl 0952.53023 [128] Kirichenko, VF; Rustanov, AR, Differential geometry of quasi-Sasakian manifolds, Mat. Sb., 193, 71-100, (2002) [129] Kirichenko, VF; Stepanova, LV, On the geometry of hypersurfaces of quasi-Kählerian manifolds, Usp. Mat. Nauk, 2, 213-214, (1995) [130] Kiritchenko, VF, Sur la gèomètrie des variètès approximativement cosymplectiques, C. R. Acad. Sci. Paris, Ser. 1, 295, 673-676, (1982) · Zbl 0519.53032 [131] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. 1, Interscience Publ., New York-London (1963). · Zbl 0119.37502 [132] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. 2, Interscience Publ., New York-London (1969). · Zbl 0175.48504 [133] Kon, M, Invariant submanifolds in Sasakian manifolds, Math. Ann., 219, 277-290, (1976) · Zbl 0301.53031 [134] Kon, M, Hypersurfaces with almost complex structures in the real affine space, Colloq. Math., 108, 329-338, (2007) · Zbl 1121.53009 [135] Korpinar, T; Turhan, E, Weierstrass formula for minimal surface in the special three-dimensional Kenmotsu manifold with $$η$$-parallel Ricci tensor, Int. J. Math. Combin., 4, 62-69, (2010) · Zbl 1227.53072 [136] Kurihara, H, On real hypersurfaces in a complex space form, Math. J. Okayama Univ., 40, 177-186, (1998) · Zbl 0951.53041 [137] Kurihara, H, The type number on real hypersurfaces in a quaternionic space form, Tsukuba J. Math., 24, 127-132, (2000) · Zbl 1034.53061 [138] Kurihara, H; Takagi, R, A note on the type number of real hypersurfaces in $$P$$_{$$n$$}($$C$$), Tsukuba J. Math., 22, 793-802, (1998) [139] G. F. Laptev, “Fundamental higher-order 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 [140] 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 [141] Li, H; Wei, G, Classification of Lagrangian Willmore submanifolds of the nearly Kähler 6-sphere $$S$$6 1 with constant scalar curvature, Glasgow Math. J., 48, 53-64, (2006) · Zbl 1117.53021 [142] Manev, M, Classes of real isotropic hypersurfaces of a Kähler manifold with $$B$$-metric, C. R. Acad. Bulgare Sci., 55, 27-32, (2002) · Zbl 1017.53054 [143] Manev, M, Classes of real time-like hypersurfaces of a Kaehler manifold with $$B$$-metric, J. Geom., 75, 113-122, (2002) · Zbl 1042.53055 [144] Matsumoto, K; Mihai, I, Warped product submanifolds in Sasakian space forms, SUT J. Math., 38, 135-144, (2002) · Zbl 1040.53074 [145] Matsumoto, K; Mihai, I; Oiaga, A, Ricci curvature of submanifolds in complex space forms, Rev. Roumaine Math. Pures Appl., 46, 775-782, (2002) · Zbl 1034.53047 [146] Matsumoto, K; Mihai, I; Rosca, R, A certain locally conformal almost cosymplectic manifold and its submanifolds, Tensor (N.S.), 64, 295-296, (2003) [147] Matzeu, P; Munteanu, M-I, Vector cross products and almost contact structures, Rend. Mat. Appl., 22, 359-376, (2002) · Zbl 1051.53023 [148] Mihai, I, Ricci curvature of submanifolds in Sasakian space forms, J. Austr. Math. Soc., 72, 247-256, (2002) · Zbl 1017.53052 [149] A. S. Mishchenko and A. T. Fomenko, A Course in Differential Geometry and Topology [in Russian], Moscow (1980). · Zbl 0524.53001 [150] Mishra, RS, Almost complex and almost contact submanifolds, Tensor (N.S.), 51, 91-102, (1992) [151] Mishra, RS, Normality of the hypersurfaces of almost Hermite manifolds, J. Indian Math. Soc., 61, 71-79, (1995) · Zbl 0857.53014 [152] Moroianu, A, Spin $$C$$-manifolds and complex contact structures, Commun. Math. Phys., 193, 661-674, (1998) · Zbl 0908.53024 [153] Nakayama, S, On a classification of almost contact metric structures, Tensor (N.S.), 9, 1-7, (1968) · Zbl 0156.42603 [154] Nannicini, A, Twistor methods in conformal almost symplectic geometry, Rend. Inst. Mat. Univ. Trieste, 34, 215-234, (2002) · Zbl 1058.53040 [155] A. P. Norden, Theory of Surfaces [in Russian], Moscow (1956). · Zbl 1148.53034 [156] Okumura, K, Odd-dimensional riemannian submanifolds admitting the almost contact metric structure in a Euclidean sphere, Tsukuba J. Math., 34, 117-128, (2010) · Zbl 1198.53016 [157] Okumura, M, Some remarks on spaces with a certain contact structure, Tôhoku Math. J., 14, 135-145, (1962) · Zbl 0119.37701 [158] Okumura, M, Certain almost contact hypersurfaces in Euclidean spaces, Kodai Math. Semin. Repts., 16, 44-54, (1964) · Zbl 0116.38901 [159] Okumura, M, Certain almost contact hypersurfaces in Kählerian manifolds of constant holomorphic sectional curvature, Tôhoku Math. J., 16, 270-284, (1964) · Zbl 0126.38101 [160] Okumura, M, Totally umbilical submanifolds of a Kählerian manifold, J. Math. Soc. Jpn., 19, 317-327, (1967) · Zbl 0166.17801 [161] Olszak, Z, On contact metric manifolds, Tôhoku Math. J., 31, 247-253, (1979) · Zbl 0397.53026 [162] G. Pitis, Geometry of Kenmotsu Manifolds, Publ. House Transilvania Univ., Brasov (2007). · Zbl 1129.53001 [163] M. M. Postnikov, Lectures in Geometry. Semester IV. Differential Geometry [in Russian], Nauka, Moscow (1988). · Zbl 0659.53001 [164] Prasad, R; Tripathi, MM, Transversal hypersurfaces of Kenmotsu manifold, Indian J. Pure Appl. Math., 34, 443-452, (2003) · Zbl 1044.53020 [165] P. K. Rashevskii, Riemannian Geometry and Tensor Analysis [in Russian], Nauka, Moscow (1967). · Zbl 0186.06502 [166] S. Sasaki, Almost Contact Manifolds, Lect. Notes, 1 (1965); 2 (1967); 3 (1968). [167] Sasaki, S; Hatakeyama, Y, On differentiable manifolds with certain structures which are closely related to almost contact structures, II, Tôhoku Math. J., 13, 281-294, (1961) · Zbl 0112.14002 [168] Sasaki, S; Hatakeyama, Y, On differentiable manifolds with contact metric structures, J. Math. Soc. Jpn., 14, 249-271, (1962) · Zbl 0109.40504 [169] Sawaki, S; Sekigawa, K, Almost Hermitian manifolds with constant holomorphic sectional curvature, J. Differ. Geom., 9, 123-134, (1974) · Zbl 0277.53036 [170] Shaikh, AA; Hui, SK, On extended generalized $$ϕ$$-recurrent $$β$$-Kenmotsu manifolds, Publ. Inst. Math., 89, 77-88, (2011) · Zbl 1289.53080 [171] Sharma, R, Contact hypersurfaces of Kähler manifolds, J. Geom., 78, 156-167, (2003) · Zbl 1053.53040 [172] 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. [173] Shukla, SS; Rao, PK, On Ricci curvature of certain submanifolds in generalized complex space forms, Tensor (N.S.), 72, 1-11, (2010) · Zbl 1226.53016 [174] Shukla, SS; Shukla, MK, On $$ϕ$$-Ricci symmetric Kenmotsu manifolds, Novi Sad J. Math., 39, 89-95, (2009) · Zbl 1224.53063 [175] L. V. Stepanova, “A quasi-Sasakian structure on hypersurfaces of Hermitian manifolds,” in: Proc. Moscow Pedagogical Inst. [in Russian], Moscow (1995), pp. 187-191. [176] L. V. Stepanova and M. B. Banaru, “On hypersurfaces of quasi-Kählerian manifolds,” in: Differential Geometry of Manifolds of Figures [in Russian], 31, Kaliningrad State Univ., Kaliningrad (2000), pp. 85-88. · Zbl 1331.53034 [177] L. V. Stepanova and M. B. Banaru, “On quasi-Sasakian and cosymplectic hypersurfaces of special Hermitian manifolds,” Differential Geometry of Manifolds of Figures [in Russian], 32, Kaliningrad State Univ., Kaliningrad (2001), pp. 87-93. · Zbl 1054.53043 [178] Stepanova, L; Banaru, M, On hypersurfaces of quasi-Kählerian manifolds, An. Ştinţ. Univ. Al. I. Cuza, Iaşi, 47, 165-170, (2001) · Zbl 1062.53050 [179] L. Stepanova and M. Banaru, “Some remarks on almost contact metric structures on hypersurfaces of a QK-manifold,” in: Webs and Quasigroups [in Russian], Tver State Univ., Tver (2002), pp. 92-96. · Zbl 1081.53519 [180] Takagi, R, On homogeneous real hypersurfaces in a complex projective space, Osaka J. Math., 10, 495-506, (1973) · Zbl 0274.53062 [181] Takagi, R, Real hypersurfaces in a complex projective space with constant principal curvatures, I, II, J. Math. Soc. Jpn., 27, 507-516, (1975) · Zbl 0311.53064 [182] Takagi, R, A class of hypersurfaces with constant principal curvatures in a sphere, J. Differ. Geom., 11, 225-233, (1976) · Zbl 0337.53003 [183] Takagi, R; Kim, I-B; Kim, BH, The rigidity for real hypersurfaces in a complex projective space, Tôhoku Math. J., 50, 531-536, (1998) · Zbl 0931.53025 [184] Takahashi, T, Sasakian $$ϕ$$-symmetric spaces, Tôhoku Math. J., 29, 91-113, (1977) · Zbl 0343.53030 [185] Tanno, S, Sasakian manifolds with constant φ-holomorphic sectional curvature, Tˆohoku Math. J., 21, 501-507, (1969) · Zbl 0188.26801 [186] Tanno, S, On the isometry groups of Sasakian manifolds, J. Math. Soc. Jpn, 22, 579-590, (1970) · Zbl 0197.48004 [187] Tanno, S, Ricci curvature of contact Riemannian manifolds, Tôhoku Math. J., 40, 441-448, (1988) · Zbl 0655.53035 [188] Tashiro, Y, On contact structures of hypersurfaces in almost complex manifolds, I, Tôhoku Math. J., 15, 62-78, (1963) · Zbl 0113.37204 [189] Tashiro, Y, On contact structures of tangent sphere bundles, Tôhoku Math. J., 21, 117-143, (1969) · Zbl 0182.55501 [190] Tripathi, MM; Shukla, SS, Ricci curvature and $$k$$-Ricci curvature for submanifolds of generalized complex space forms, Aligarh Bull. Math., 20, 143-156, (2001) · Zbl 1079.40502 [191] Venkatesha; Bagewadi, CS, Some curvature tensors on a Kenmotsu manifold, Tensor (N.S.), 68, 140-147, (2007) · Zbl 1193.53117 [192] Voicu, RC, Ricci curvature properties and stability on 3-dimensional Kenmotsu manifolds, Contemp. Math., 542, 273-278, (2011) · Zbl 1222.53069 [193] Wu, B, 1-type minimal surfaces in complex Grassmann manifolds and its Gauss map, Tsukuba J. Math., 26, 49-60, (2002) · Zbl 1025.53036 [194] K. Yano, Differential Geometry on Complex and Almost Complex Spaces, Pergamon Press, Oxford (1965). · Zbl 0127.12405 [195] 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 [196] K. Yano and M. Kon, Structures on Manifolds, World Scientific, Singapore (1984). · Zbl 0557.53001 [197] Yoon Dae Won, “Inequality for Ricci curvature of certain submanifolds in locally conformal almost cosymplectic manifolds,” Int. J. Math. Math. Sci., 10, 1621-1632 (2005). · Zbl 1089.53041 [198] Zhang, X, Energy properness and Sasakian-Einstein metrics, Commun. Math. Phys., 306, 229-260, (2011) · Zbl 1226.53049 [199] Tongde Zhong, “The geometry of hypersurfaces in a Kähler manifold,” Acta Math. Sci., Ser. B, 21, 350-362 (2001). · Zbl 0999.53042 [200] Zhenrong Zhou, “Spectral geometry of Sasakian submanifolds,” J. Math. (P.R. China), 20, No. 1, 83-86 (2000). · Zbl 0958.58008
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.