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Year 2023, , 464 - 525, 29.10.2023
https://doi.org/10.36890/iejg.1300339

Abstract

References

  • [1] Acet, B. E., Perkta¸s, S. Y., Kılıç, E.: Kenmotsu manifolds with generalized Tanaka-Webster connection. Adıyaman Üniversitesi Fen Bilimleri Dergisi 3 (2) 79-93 (2013). http://dspace2.adiyaman.edu.tr:8080/xmlui/handle/20.500.12414/330
  • [2] Adamów, A., Deszcz, R.: On totally umbilical submanifolds of some class Riemannian manifolds. Demonstratio Math. 16 (1), 39-59 (1983). https://doi.org/10.1515/dema-1983-0105
  • [3] Aktan, M., Yildirim, Murathan, M. C.: Almost f-cosymplectic manifolds, Mediterr. J. Math. 11 (2), 775-787 (2014). https://doi.org/10.1007/s00009-013-0329-2
  • [4] Bishop, R. L., O’Neill, B.: Manifolds of negative curvature. Trans. Amer. Math. Soc. 145 1-49 (1969). https://doi.org/10.1090/S0002-9947- 1969-0251664-4
  • [5] Blair, D. E.: Almost contact manifolds with Killing structure tensors. Pacific J. Math. 39 (3) 285-292 (1971).
  • [6] Blair, D. E.: Riemannian Geometry of Contact and Symplectic Manifolds, second edition, Progress in Mathematics. Birkhäuser Verlag, Boston (2010).
  • [7] Blair, D. E., T. Koufogiorgos, T., Sharma, R.: A classification of three-dimensional contact metric manifolds with Qφ = φQ. Kodai Math. J. 13 (3) 391-401 (1990). https://doi.org/10.2996/kmj/1138039284
  • [8] Boeckx, E., Buken, P., Vanhecke, L.: φ-symmetric contact metric spaces. Glasgow Math. J. 41 (3) 409-416 (1999). https://doi.org/10.1017/S0017089599000579
  • [9] Boeckx, E., Kowalski, O., Vanhecke, V.: Riemannian Manifolds of Conullity Two, World Scientific, Singapore (1996).
  • [10] Boeckx, E., Vanhecke, L.: Characteristic reflections on unit tangent sphere bundles. Houston J. Math. 23 427-448 (1997).
  • [11] Buken, P., Vanhecke, L.: Reflections in K-contact geometry. Math. Rep. Toyama Univ. 12 41-49 (1989).
  • [12] Boyer, C. P., Galicki, K.: Sasakian Geometry, Oxford Mathematical Monographs, Oxford University Press, Oxford (2008).
  • [13] Calvaruso, G.: Conformally flat semi-symmetric spaces. Archiv. Math. (Brno) 41 (1) 27-36 (2005). https://dml.cz/dmlcz/107933
  • [14] Calvaruso, G.: Conformally flat pseudo-symmetric spaces of constant type. Czech. Math. J. 56 (2006), (2) 649-657 (2006). https://dml.cz/dmlcz/128094
  • [15] Calvaruso, G., Perrone, A.: Natural almost contact structures and their 3D homogeneous models. Math. Nachr. 289 (11-12) 1370-1385 (2016). https://doi.org/10.1002/mana.201400315
  • [16] Carriazo, A., Martín-Molina, V.: Almost Cosymplectic and almost Kenmotsu (κ, μ, ν)-spaces. Mediterr. J. Math. 10, 1551-1571 (2013). https://doi.org/10.1007/s00009-013-0246-4
  • [17] Cartan, E.: Leçons sur la géométrie des espaces de Riemann 2nd. ed., Paris, 1946.
  • [18] Cheng, Q.-M., Ishikawa, S., Shiohama, K.: Conformally flat 3-manifolds with constant scalar curvature. J. Math. Soc. Japan 51 (1), 209-226 (1999). https://doi.org/10.2969/jmsj/05110209
  • [19] Chinea, D.: Harmonicity on maps between almost contact metric manifolds. Acta Math. Hungar. 126 (4), 352-362 (2010).https://doi.org/10.1007/s10474-009-9076-z
  • [20] Cho, J. T.: On some classes of almost contact metric manifolds. Tsukuba J. Math. 19 (1) 201-217 (1995). https://www.jstor.org/stable/43685920
  • [21] Cho, J. T.: Notes on almost Kenmotsu three-manifolds. Honam Math. J. 36 (3) 637-645 (2014). https://doi.org/10.5831/HMJ.2014.36.3.637
  • [22] Cho, J. T.: Local symmetry on almost Kenmotsu three-manifolds. Hokkaido Math. J. 45 (2016) 435-442 (2016). https://doi.org/10.14492/hokmj/1478487619
  • [23] Cho, J. T.: Notes in real hypersurfaces in a complex space form. Bull. Korean Math. Soc. 52 (1) 335-344 (2015). https://doi.org/10.4134/BKMS.2015.52.1.335
  • [24] Cho, J. T., Inoguchi, J.: On φ-Einstein contact Riemannian manifolds. Meditterr. J. Math. 7 (2) 143-167 (2010). https://doi.org/10.1007/s00009-010-0049-9
  • [25] Cho, J. T., Inoguchi, J.: Characteristic Jacobi operator on contact Riemannian 3-manifolds. Differ. Geom. Dyn. Syst. 17 49-71 (2015). http://www.mathem.pub.ro/dgds/v17/D17-cj-987.pdf
  • [26] Cho, J. T., Inoguchi, J.: Contact 3-manifolds with Reeb flow invariant characteristic Jacobi operator. An. Ştiinț. Univ. Al. I. Cuza Mat. N. S. 63 (3) 665-676 (2017).
  • [27] Cho, J. T., Kimura, M.: Reeb flow symmetry on almost contact three-manifolds. Differential Geom. Appl. 35 suppl. 266-273 (2014). https://doi.org/10.1016/j.difgeo.2014.05.002
  • [28] Deszcz, R., Grycak, W.: On some class of warped product manifolds. Bull. Inst. Math. Acad. Sinica 15 (3) 311-322 (1987).
  • [29] De, U. C.: On Φ-symmetric Kenmotsu manifolds. Int. Electron. J. Geom. 1 (1) 33-38 (2008). https://dergipark.org.tr/en/pub/iejg/issue/46277/581499
  • [30] De. U. C., Pathak, G.: On 3-dimensional Kenmotsu manifolds. Indian J. Pure Applied Math. 35 159-165 (2004).
  • [31] Dey, D., Majhi, P.: ∗-Ricci tensor on almost Kenmotsu 3-manifolds. Int. J. Geom. Methods Mod. Phys. 2020 2050196 (11 pages) (2020)https://doi.org/10.1142/S0219887820501960
  • [32] Dileo, G.: On the geometry of almost contact metric manifolds of Kenmotsu type. Differ. Geom. Appl. 29 Suppl. 1 S58-S64 (2011). https://doi.org/10.1016/j.difgeo.2011.04.008
  • [33] Dileo, G.: A classification of certain almost α-Kenmotsu manifolds. Kodai Math. J. 34 (3), 426-445 (2011). https://doi.org/10.2996/kmj/1320935551
  • [34] Dileo, G., Pastore, A. M.: Almost Kenmotsu manifolds and local symmetry. Bull. Belg. Math. Soc. Simon Stevin 14 343-354 (2007). https://doi.org/10.36045/bbms/1179839227
  • [35] Dileo, G., Pastore, A. M.: Almost Kenmotsu manifolds with a condition of η-parallelism. Differ. Geom. Appl. 27, (5) 671-679 (2009) 671–679 https://doi.org/10.1016/j.difgeo.2009.03.007
  • [36] Dileo, G., Pastore, A. M.: Almost Kenmotsu manifolds and nullity distributions. J. Geom. 93 46-61 (2009). https://doi.org/10.1007/s00022-009-1974-2
  • [37] Dragomir, S., Kamishima, Y.: Pseudoharmonic maps and vector fields on CR manifolds. J. Math. Soc. Japan 62 (1), 269-303 (2010). https://doi.org/10.2969/jmsj/06210269
  • [38] Dragomir, S., Perrone, D.: Harmonic Vector Fields: Variational Principles and Differential Geometry, Elsevier, (2012).
  • [39] Dragomir, S., Perrone, D.: Levi harmonic maps of contact Riemannian manifolds. J. Geom. Anal. 24 (3) 1233-1275 (2014). https://doi.org/10.1007/s12220-012-9371-8
  • [40] Erdem, S.: On harmonicity of holomorphic maps between various types of almost contact metric manifolds. (2023). https://doi.org/10.48550/arXiv.2302.12677
  • [41] Gherghe, C.: Harmonic maps on Kenmotsu manifolds. Rev. Roumaine Math. Pure Appl. 45 447-453 (2000).
  • [42] Gherghe, C., Vîlcu, G. E.: Harmonic maps on locally conformal almost cosymplectic manifolds, submitted.
  • [43] Góes, C. C., Simões, P. A.: The generalized Gauss map of minimal surfaces in H3 and H4. Bol. Soc. Brasil Mat. 18 35-47 (1987). https://doi.org/10.1007/BF02590022
  • [44] Ghosh, G., De, U. C.: Kenmotsu manifolds with generalized Tanaka-Webster connection. Publ. Inst. Math. (Beograd) (N.S.) 102 (116) 221–230 (2017). https://doi.org/10.2298/PIM1716221G
  • [45] Haesen, S., Verstraelen, L.: On the sectional curvature of Deszcz, An. ¸Stiin¸t. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 53 supp., 181-190 (2007).
  • [46] Haesen, S., Verstraelen, L.: Natural intrinsic geometrical symmetries, SIGMA 5 Article Number 086, 15 pages (2009).
  • [47] Inoguchi, J.: Minimal surfaces in 3-dimensional solvable Lie groups. Chinese Ann. Math. B. 24 73-84 (2003). https://doi.org/10.1142/S0252959903000086
  • [48] Inoguchi, J.: Minimal surfaces in 3-dimensional solvable Lie groups II. Bull. Austral. Math. Soc. 73 365-374 (2006). https://doi.org/10.1017/S0004972700035401
  • [49] Inoguchi, J.: Pseudo-symmetric Lie groups of dimension 3. Bull. Fac. Edu. Utsunomiya Univ. Sect. 2 57 1-5 (2007). http://hdl.handle.net/10241/00004810
  • [50] Inoguchi, J.: On homogenous contact 3-manifolds. Bull. Fac. Edu. Utsunomiya Univ. Sect. 2 59 1-12 (2009). http://hdl.handle.net/10241/00004788
  • [51] Inoguchi, J.: A note on almost contact Riemannian 3-manifolds. Bull. Yamagata Univ. Natur. Sci. 17 (1) 1-6 (2010) http://id.nii.ac.jp/1348/00003041/
  • [52] Inoguchi, J.: Harmonic maps in almost contact geometry. SUT J. Math. 50 (2) 353-382 (2014) http://doi.org/10.20604/00000831
  • [53] Inoguchi, J.: A note on almost contact Riemannian 3-manifolds II. Bull. Korean Math. Soc. 54 (1) 85-97 (2017). https://doi.org/10.4134/BKMS.b150772
  • [54] Inoguchi, J.: Characteristic Jacobi operator on almost cosymplectic 3-manifolds. Int. Electron. J. Geom. 12 (2) 276-299 (2019). https://doi.org/10.36890/iejg.584487
  • [55] Inoguchi, J.: J-trajectories in locally conformal Kähler manifolds with parallel anti Lee field. Int. Electron. J. Geom. 13 (2) 30-44 (2020). https://doi.org/10.36890/iejg.718806
  • [56] Inoguchi, J.: On some curves in 3-dimensional hyperbolic geometry and solvgeometry. J. Geom. 113, Article number: 37 (2022). https://doi.org/10.1007/s00022-022-00650-6
  • [57] Inoguchi, J.: Homogeneous Riemannian structures in Thurston geometries and contact Riemannian geometries, in preparation.
  • [58] Inoguchi, J., Lee, J.-E.: Affine biharmonic curves in 3-dimensional homogeneous geometries. Mediterr. J. Math. 10 (1) 571-592 (2013). https://doi.org/10.1007/s00009-012-0195-3
  • [59] Inoguchi, J., Lee, J.-E.: Slant curves in 3-dimensional almost contact metric geometry. Int. Electron. J. Geom. 8 (2) 106-146 (2015). https://doi.org/10.36890/iejg.592300
  • [60] Inoguchi, J., Lee, J.-E.: Slant curves in 3-dimensional almost f-Kenmotsu manifolds. Comm. Korean Math. Soc. 32 (2) 417-424 (2017). https://doi.org/10.4134/CKMS.c160079
  • [61] Inoguchi, J., Lee, J.-E.: Biharmonic curves in f-Kenmotsu 3-manifolds. J. Math. Anal. Appl. 509 (1) 125942 (2022). https://doi.org/10.1016/j.jmaa.2021.125941
  • [62] Inoguchi, J., Lee, J.-E.: φ-trajectories in Kenmotsu manifolds. J. Geom. 113 (1) 8 (2022). https://doi.org/10.1007/s00022-021-00624-0
  • [63] Inoguchi, J., Lee, J.-E.: Almost Kenmotsu 3-manifolds with pseudo-parallell characteristic Jacobi operator. Results Math. 78 Article Number 48 (2023). https://doi.org/10.1007
  • [64] Inoguchi, J., Lee, J.-E.: Pseudo-symmetric almost Kenmotsu 3-manifolds, submitted.
  • [65] Inoguchi, J., Lee, J.-E.: On the η-parallelism in almost Kenmotsu 3-manifolds. J. Korean Math. Soc., to appear.
  • [66] Inoguchi, J., Lee, S.: A Weierstrass representation for minimal surfaces in Sol. Proc. Amer. Math. Soc. 136, 2209-2216 (2008). https://doi.org/10.1090/S0002-9939-08-09161-2
  • [67] Inoguchi, J., Naitoh, H.: Grassmann geometry on the 3-dimensional non-unimodular Lie groups. Hokkaido Math. J. 48 (2): 385-406 (2019). https://doi.org/10.14492/hokmj/1562810516
  • [68] Janssens, D., Vanhecke, L.: Almost contact structures and curvature tensors. Kodai Math. J. 4 (1), 1-27 (1981). https://doi.org/10.2996/kmj/1138036310
  • [69] Kenmotsu, K.: A class of almost contact Riemannian manifolds. Tohoku Math. J. 24, 93-103 (1972) https://doi.org/10.2748/tmj/1178241594
  • [70] Kenmotsu, K.: Weierstrass formula for surfaces of prescribed mean curvature. Math. Ann. 245, 89-99 (1979). https://doi.org/10.1007/BF01428799
  • [71] Kim, T. W., Pak, H. K.: Canonical foliations of certain classes of almost contact metric structures. Acta Math. Sin. (Engl. Ser.) 21 (4), 841-846 (2005). https://doi.org/10.1007/s10114-004-0520-2
  • [72] Kimura, M., Maeda, S.: On real hypersurfaces of a complex projective space. Math. Z. 202 (3) 299-311 (1989). https://doi.org/10.1007/BF01159962
  • [73] Kiran Kumar, D. L., Nagaraja, H. G., Manjulamma, U., Shashidhar, S.: Study on Kenmotsu manifolds admitting generalized Tanaka-Webster connection. Ital. J. Pure Appl. Math. 47 721-733 (2022).
  • [74] Kokubu, M.: Weierstrass representation for minimal surfaces in hyperbolic space. Tohoku Math. J. 49 (3) 367–377 (1997). https://doi.org/10.2748/tmj/1178225110
  • [75] Kon, M.: Invariant submanifolds in Sasakian manifolds. Math. Ann. 219 (3) 277-290 (1976). https://doi.org/10.1007/BF01354288
  • [76] Kowalski, O.: An explicit classication of 3-dimensional Riemannian spaces satisfying R(X, Y ) · R = 0, Czech. Math. J. 46 (3) 427-474 (1996).
  • [77] Kowalski, O., Sekizawa, M.: Pseudo-symmetric spaces of constant type in dimension three–elliptic spaces. Rend. Mat. VII. 17 477-512 (1997).
  • [78] Milnor, J.: Curvatures of left invariant metrics on Lie groups. Adv. Math. 21 (3) 293-329 (1976). https://doi.org/10.1016/S0001-8708(76)80002-3
  • [79] Nistor, A.I.: Constant angle surfaces in solvable Lie groups. Kyushu J. Math. 68 (2) 315-332 (2014) http://dx.doi.org/10.2206/kyushujm.68.315
  • [80] Okumura, M.: Some remarks on space with a certain contact structure. Tôhoku Math. 14 135-145 (1962). https://doi.org/10.2748/tmj/1178244168
  • [81] Olszak, Z.: On almost cosymplectic manifolds. Kodai Math. J. 4 (2) 239-250 (1981). https://doi.org/10.2996/kmj/1138036371
  • [82] Olszak, Z.: Normal almost contact metric manifolds of dimension three. Ann. Pol. Math. 47 41-50 (1986). https://doi.org/10.4064/ap-47-1-41-50
  • [83] O’Neill, B.: Semi-Riemannian Geometry with Application to Relativity. Academic Press, Orland, 1983.
  • [84] Öztürk, H.: On almost alpha-cosymplectic manifolds with some nullity distributions. Honam Math. J. 41 (2), 269–284 (2019). https://doi.org/10.5831/HMJ.2019.41.2.269
  • [85] Öztürk, H., N. Aktan, N., Murathan, C.: Almost α-cosymplectic (κ, μ, ν)-spaces. Preprint arXiv:1007.0527v1 [math.DG] (2010).
  • [86] Pak, H.-K.: Canonical foliations of almost f-cosymplectic structures, J. Korea Ind. Inf. Syst. Res. 7 (3), 89-94, (2002).
  • [87] Pan, Q.,Wu, H.,Wang, Y.: Almost Kenmotsu 3-h-manifolds with transversely Killing-type Ricci operators. Open Math. 18 (1) 1056-1063 (2020). https://doi.org/10.1515/math-2020-0057
  • [88] Pastore, A. M., Saltarelli, V.: Generalized nullity conditions on almost Kenmotsu manifolds. Int. Electron. J. Geom. 4 168-183 (2011). https://dergipark.org.tr/en/pub/iejg/issue/47488/599509
  • [89] Perrone, D.: Weakly ϕ-symmetric contact metric spaces. Balkan J. Geom. Appl. 7 (2) 67-77 (2002).
  • [90] Perrone, D.: Classification of homogeneous almost cosymplectic three-manifolds. Differ. Geom. Appl. 30 (1) 49-58 (2012). https://doi.org/10.1016/j.difgeo.2011.10.003
  • [91] Perrone, D.: Almost contact metric manifolds whose Reeb vector field is a harmonic section. Acta Math. Hung. 138 (2013), 102-126. https://doi.org/10.1007/s10474-012-0228-1
  • [92] Perrone, D., Remarks on Levi harmonicity of contact semi-Riemannian manifolds. J. Korean Math. Soc. 51 (5), 881-895 (2014). http://dx.doi.org/10.4134/JKMS.2014.51.5.881
  • [93] Perrone, D.: Classification of homogeneous almost α-coKähler three-manifolds. Differ. Geom. Appl. 59 66–90 (2018) 66–90 https://doi.org/10.1016/j.difgeo.2011.10.00
  • [94] Perrone, D.: Left-invariant almost α-coKähler structures on 3D semidirect product Lie groups. Int. J. Geom. Methods Mod. Phys. 16 (1) 1950011 18 pages (2019). https://doi.org/10.1142/S0219887819500117
  • [95] Perrone, D., Almost contact Riemannian three-manifolds with Reeb flow symmetry. Differ. Geom. Appl. 75 (2021), Article ID 101736 11 pages(2021). https://doi.org/10.1016/j.difgeo.2021.101736
  • [96] Rehman, N. A.: Harmonic maps on Kenmotsu manifolds. An. S¸t. Univ. Ovidius Constanta Ser. Math. 21 (3) 197-208 (2013).
  • [97] Saltarelli, V., Three-dimensional almost Kenmotsu manifolds satisfying certain nullity conditions. Bull. Malays. Math. Sci. Soc. 38, 437-459(2015). https://doi.org/10.1007/s40840-014-0029-5
  • [98] Sasaki, S., Hatakeyama, Y.: On differentiable manifolds with certain structures which are close related to almost contact structure II. Tôhoku Math. 13 281-294 (1961). https://doi.org/10.2748/tmj/1178244304
  • [99] Sekigawa, K.: Some 3-dimensional Riemannian manifolds with constant scalar curvature, Sci. Rep. Niigata Univ. Ser. A. 11 25-29 (1974).
  • [100] Szabó, Z. I.: Structure theorems on Riemannian manifolds satisfying R(X, Y ) · R = 0, I. the local version, J. Differ. Geom. 17 531-582 (1982).
  • [101] Tachibana, S.: A theorem on Riemannian manifolds of positive curvature operator. Proc. Japan Acad. 50, 301-302 (1974). https://doi.org/10.3792/pja/1195518988
  • [102] Takahashi, T.: Sasakian ϕ-symmetric spaces. Tohoku Math. J. (2) 29 (1) 91-113 (1977). https://doi.org/10.2748/tmj/1178240699
  • [103] Tanno, S.: The automorphism groups of almost contact Riemannian manifolds. Tohoku Math. J. (2) 21 (1), 21-38 (1969). https://doi.org/10.2748/tmj/1178243031
  • [104] Tanno, S.: Variational problems on contact Riemannian manifolds. Trans. Amer. Math. Soc. 314 349-379 (1989). https://doi.org/10.1090/S0002-9947-1989-1000553-9
  • [105] Tasaki, H., Umehara, M.: An invariant on 3-dimensional Lie algebras. Proc. Amer. Math. Soc. 115 (2) 293-294 (1992). https://doi.org/10.1090/S0002-9939-1992-1087471-0
  • [106] Tricerri, F. and Vanhecke, L., Homogeneous Structures on Riemannian Manifolds, Lecture Notes Series, London Math. Soc. 52, (1983), Cambridge Univ. Press.
  • [107] Vaisman, I.: Conformal changes of almost contact metric structures, in: R. Artzy, I. Vaisman (Eds.), Geometry and Differential Geometry (Artzy, R., Vaisman, I., Eds.), Lecture Notes in Math. 792, Springer Verlag, 435–44, (1980).
  • [108] Voicu, R. C.: Ricci curvature properties and stability on 3-dimensional Kenmotsu manifolds. Harmonic maps and differential geometry, 273- 278, Contemp. Math., 542, Amer. Math. Soc., Providence, RI, 2011.
  • [109] Wang, Y.: Three-dimensional locally symmetric almost Kenmotsu manifolds, Ann. Pol. Math. 116 (1), 79-86 (2016). https://doi.org/10.4064/ap3555-12-2015
  • [110] Wang, Y.: A class of 3-dimensional almost Kenmotsu manifolds with harmonic curvature tensors. Open Math. 14, 977-985 (2016).
  • [111] Wang, Y., Liu, X.: Locally symmetric CR-integrable almost Kenmotsu manifolds, Mediterr. J. Math. 12 159-171 (2015). https://doi.org/10.1007/s00009-014-0388-z

Characteristic Jacobi Operator on Almost Kenmotsu $3$-manifolds

Year 2023, , 464 - 525, 29.10.2023
https://doi.org/10.36890/iejg.1300339

Abstract

The Ricci tensor field, $\varphi$-Ricci tensor field and the characteristic Jacobi operator
on almost Kenmotsu $3$-manifolds are investigated. We give a classification of
locally symmetric almost Kenmotsu $3$-manifolds.

References

  • [1] Acet, B. E., Perkta¸s, S. Y., Kılıç, E.: Kenmotsu manifolds with generalized Tanaka-Webster connection. Adıyaman Üniversitesi Fen Bilimleri Dergisi 3 (2) 79-93 (2013). http://dspace2.adiyaman.edu.tr:8080/xmlui/handle/20.500.12414/330
  • [2] Adamów, A., Deszcz, R.: On totally umbilical submanifolds of some class Riemannian manifolds. Demonstratio Math. 16 (1), 39-59 (1983). https://doi.org/10.1515/dema-1983-0105
  • [3] Aktan, M., Yildirim, Murathan, M. C.: Almost f-cosymplectic manifolds, Mediterr. J. Math. 11 (2), 775-787 (2014). https://doi.org/10.1007/s00009-013-0329-2
  • [4] Bishop, R. L., O’Neill, B.: Manifolds of negative curvature. Trans. Amer. Math. Soc. 145 1-49 (1969). https://doi.org/10.1090/S0002-9947- 1969-0251664-4
  • [5] Blair, D. E.: Almost contact manifolds with Killing structure tensors. Pacific J. Math. 39 (3) 285-292 (1971).
  • [6] Blair, D. E.: Riemannian Geometry of Contact and Symplectic Manifolds, second edition, Progress in Mathematics. Birkhäuser Verlag, Boston (2010).
  • [7] Blair, D. E., T. Koufogiorgos, T., Sharma, R.: A classification of three-dimensional contact metric manifolds with Qφ = φQ. Kodai Math. J. 13 (3) 391-401 (1990). https://doi.org/10.2996/kmj/1138039284
  • [8] Boeckx, E., Buken, P., Vanhecke, L.: φ-symmetric contact metric spaces. Glasgow Math. J. 41 (3) 409-416 (1999). https://doi.org/10.1017/S0017089599000579
  • [9] Boeckx, E., Kowalski, O., Vanhecke, V.: Riemannian Manifolds of Conullity Two, World Scientific, Singapore (1996).
  • [10] Boeckx, E., Vanhecke, L.: Characteristic reflections on unit tangent sphere bundles. Houston J. Math. 23 427-448 (1997).
  • [11] Buken, P., Vanhecke, L.: Reflections in K-contact geometry. Math. Rep. Toyama Univ. 12 41-49 (1989).
  • [12] Boyer, C. P., Galicki, K.: Sasakian Geometry, Oxford Mathematical Monographs, Oxford University Press, Oxford (2008).
  • [13] Calvaruso, G.: Conformally flat semi-symmetric spaces. Archiv. Math. (Brno) 41 (1) 27-36 (2005). https://dml.cz/dmlcz/107933
  • [14] Calvaruso, G.: Conformally flat pseudo-symmetric spaces of constant type. Czech. Math. J. 56 (2006), (2) 649-657 (2006). https://dml.cz/dmlcz/128094
  • [15] Calvaruso, G., Perrone, A.: Natural almost contact structures and their 3D homogeneous models. Math. Nachr. 289 (11-12) 1370-1385 (2016). https://doi.org/10.1002/mana.201400315
  • [16] Carriazo, A., Martín-Molina, V.: Almost Cosymplectic and almost Kenmotsu (κ, μ, ν)-spaces. Mediterr. J. Math. 10, 1551-1571 (2013). https://doi.org/10.1007/s00009-013-0246-4
  • [17] Cartan, E.: Leçons sur la géométrie des espaces de Riemann 2nd. ed., Paris, 1946.
  • [18] Cheng, Q.-M., Ishikawa, S., Shiohama, K.: Conformally flat 3-manifolds with constant scalar curvature. J. Math. Soc. Japan 51 (1), 209-226 (1999). https://doi.org/10.2969/jmsj/05110209
  • [19] Chinea, D.: Harmonicity on maps between almost contact metric manifolds. Acta Math. Hungar. 126 (4), 352-362 (2010).https://doi.org/10.1007/s10474-009-9076-z
  • [20] Cho, J. T.: On some classes of almost contact metric manifolds. Tsukuba J. Math. 19 (1) 201-217 (1995). https://www.jstor.org/stable/43685920
  • [21] Cho, J. T.: Notes on almost Kenmotsu three-manifolds. Honam Math. J. 36 (3) 637-645 (2014). https://doi.org/10.5831/HMJ.2014.36.3.637
  • [22] Cho, J. T.: Local symmetry on almost Kenmotsu three-manifolds. Hokkaido Math. J. 45 (2016) 435-442 (2016). https://doi.org/10.14492/hokmj/1478487619
  • [23] Cho, J. T.: Notes in real hypersurfaces in a complex space form. Bull. Korean Math. Soc. 52 (1) 335-344 (2015). https://doi.org/10.4134/BKMS.2015.52.1.335
  • [24] Cho, J. T., Inoguchi, J.: On φ-Einstein contact Riemannian manifolds. Meditterr. J. Math. 7 (2) 143-167 (2010). https://doi.org/10.1007/s00009-010-0049-9
  • [25] Cho, J. T., Inoguchi, J.: Characteristic Jacobi operator on contact Riemannian 3-manifolds. Differ. Geom. Dyn. Syst. 17 49-71 (2015). http://www.mathem.pub.ro/dgds/v17/D17-cj-987.pdf
  • [26] Cho, J. T., Inoguchi, J.: Contact 3-manifolds with Reeb flow invariant characteristic Jacobi operator. An. Ştiinț. Univ. Al. I. Cuza Mat. N. S. 63 (3) 665-676 (2017).
  • [27] Cho, J. T., Kimura, M.: Reeb flow symmetry on almost contact three-manifolds. Differential Geom. Appl. 35 suppl. 266-273 (2014). https://doi.org/10.1016/j.difgeo.2014.05.002
  • [28] Deszcz, R., Grycak, W.: On some class of warped product manifolds. Bull. Inst. Math. Acad. Sinica 15 (3) 311-322 (1987).
  • [29] De, U. C.: On Φ-symmetric Kenmotsu manifolds. Int. Electron. J. Geom. 1 (1) 33-38 (2008). https://dergipark.org.tr/en/pub/iejg/issue/46277/581499
  • [30] De. U. C., Pathak, G.: On 3-dimensional Kenmotsu manifolds. Indian J. Pure Applied Math. 35 159-165 (2004).
  • [31] Dey, D., Majhi, P.: ∗-Ricci tensor on almost Kenmotsu 3-manifolds. Int. J. Geom. Methods Mod. Phys. 2020 2050196 (11 pages) (2020)https://doi.org/10.1142/S0219887820501960
  • [32] Dileo, G.: On the geometry of almost contact metric manifolds of Kenmotsu type. Differ. Geom. Appl. 29 Suppl. 1 S58-S64 (2011). https://doi.org/10.1016/j.difgeo.2011.04.008
  • [33] Dileo, G.: A classification of certain almost α-Kenmotsu manifolds. Kodai Math. J. 34 (3), 426-445 (2011). https://doi.org/10.2996/kmj/1320935551
  • [34] Dileo, G., Pastore, A. M.: Almost Kenmotsu manifolds and local symmetry. Bull. Belg. Math. Soc. Simon Stevin 14 343-354 (2007). https://doi.org/10.36045/bbms/1179839227
  • [35] Dileo, G., Pastore, A. M.: Almost Kenmotsu manifolds with a condition of η-parallelism. Differ. Geom. Appl. 27, (5) 671-679 (2009) 671–679 https://doi.org/10.1016/j.difgeo.2009.03.007
  • [36] Dileo, G., Pastore, A. M.: Almost Kenmotsu manifolds and nullity distributions. J. Geom. 93 46-61 (2009). https://doi.org/10.1007/s00022-009-1974-2
  • [37] Dragomir, S., Kamishima, Y.: Pseudoharmonic maps and vector fields on CR manifolds. J. Math. Soc. Japan 62 (1), 269-303 (2010). https://doi.org/10.2969/jmsj/06210269
  • [38] Dragomir, S., Perrone, D.: Harmonic Vector Fields: Variational Principles and Differential Geometry, Elsevier, (2012).
  • [39] Dragomir, S., Perrone, D.: Levi harmonic maps of contact Riemannian manifolds. J. Geom. Anal. 24 (3) 1233-1275 (2014). https://doi.org/10.1007/s12220-012-9371-8
  • [40] Erdem, S.: On harmonicity of holomorphic maps between various types of almost contact metric manifolds. (2023). https://doi.org/10.48550/arXiv.2302.12677
  • [41] Gherghe, C.: Harmonic maps on Kenmotsu manifolds. Rev. Roumaine Math. Pure Appl. 45 447-453 (2000).
  • [42] Gherghe, C., Vîlcu, G. E.: Harmonic maps on locally conformal almost cosymplectic manifolds, submitted.
  • [43] Góes, C. C., Simões, P. A.: The generalized Gauss map of minimal surfaces in H3 and H4. Bol. Soc. Brasil Mat. 18 35-47 (1987). https://doi.org/10.1007/BF02590022
  • [44] Ghosh, G., De, U. C.: Kenmotsu manifolds with generalized Tanaka-Webster connection. Publ. Inst. Math. (Beograd) (N.S.) 102 (116) 221–230 (2017). https://doi.org/10.2298/PIM1716221G
  • [45] Haesen, S., Verstraelen, L.: On the sectional curvature of Deszcz, An. ¸Stiin¸t. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 53 supp., 181-190 (2007).
  • [46] Haesen, S., Verstraelen, L.: Natural intrinsic geometrical symmetries, SIGMA 5 Article Number 086, 15 pages (2009).
  • [47] Inoguchi, J.: Minimal surfaces in 3-dimensional solvable Lie groups. Chinese Ann. Math. B. 24 73-84 (2003). https://doi.org/10.1142/S0252959903000086
  • [48] Inoguchi, J.: Minimal surfaces in 3-dimensional solvable Lie groups II. Bull. Austral. Math. Soc. 73 365-374 (2006). https://doi.org/10.1017/S0004972700035401
  • [49] Inoguchi, J.: Pseudo-symmetric Lie groups of dimension 3. Bull. Fac. Edu. Utsunomiya Univ. Sect. 2 57 1-5 (2007). http://hdl.handle.net/10241/00004810
  • [50] Inoguchi, J.: On homogenous contact 3-manifolds. Bull. Fac. Edu. Utsunomiya Univ. Sect. 2 59 1-12 (2009). http://hdl.handle.net/10241/00004788
  • [51] Inoguchi, J.: A note on almost contact Riemannian 3-manifolds. Bull. Yamagata Univ. Natur. Sci. 17 (1) 1-6 (2010) http://id.nii.ac.jp/1348/00003041/
  • [52] Inoguchi, J.: Harmonic maps in almost contact geometry. SUT J. Math. 50 (2) 353-382 (2014) http://doi.org/10.20604/00000831
  • [53] Inoguchi, J.: A note on almost contact Riemannian 3-manifolds II. Bull. Korean Math. Soc. 54 (1) 85-97 (2017). https://doi.org/10.4134/BKMS.b150772
  • [54] Inoguchi, J.: Characteristic Jacobi operator on almost cosymplectic 3-manifolds. Int. Electron. J. Geom. 12 (2) 276-299 (2019). https://doi.org/10.36890/iejg.584487
  • [55] Inoguchi, J.: J-trajectories in locally conformal Kähler manifolds with parallel anti Lee field. Int. Electron. J. Geom. 13 (2) 30-44 (2020). https://doi.org/10.36890/iejg.718806
  • [56] Inoguchi, J.: On some curves in 3-dimensional hyperbolic geometry and solvgeometry. J. Geom. 113, Article number: 37 (2022). https://doi.org/10.1007/s00022-022-00650-6
  • [57] Inoguchi, J.: Homogeneous Riemannian structures in Thurston geometries and contact Riemannian geometries, in preparation.
  • [58] Inoguchi, J., Lee, J.-E.: Affine biharmonic curves in 3-dimensional homogeneous geometries. Mediterr. J. Math. 10 (1) 571-592 (2013). https://doi.org/10.1007/s00009-012-0195-3
  • [59] Inoguchi, J., Lee, J.-E.: Slant curves in 3-dimensional almost contact metric geometry. Int. Electron. J. Geom. 8 (2) 106-146 (2015). https://doi.org/10.36890/iejg.592300
  • [60] Inoguchi, J., Lee, J.-E.: Slant curves in 3-dimensional almost f-Kenmotsu manifolds. Comm. Korean Math. Soc. 32 (2) 417-424 (2017). https://doi.org/10.4134/CKMS.c160079
  • [61] Inoguchi, J., Lee, J.-E.: Biharmonic curves in f-Kenmotsu 3-manifolds. J. Math. Anal. Appl. 509 (1) 125942 (2022). https://doi.org/10.1016/j.jmaa.2021.125941
  • [62] Inoguchi, J., Lee, J.-E.: φ-trajectories in Kenmotsu manifolds. J. Geom. 113 (1) 8 (2022). https://doi.org/10.1007/s00022-021-00624-0
  • [63] Inoguchi, J., Lee, J.-E.: Almost Kenmotsu 3-manifolds with pseudo-parallell characteristic Jacobi operator. Results Math. 78 Article Number 48 (2023). https://doi.org/10.1007
  • [64] Inoguchi, J., Lee, J.-E.: Pseudo-symmetric almost Kenmotsu 3-manifolds, submitted.
  • [65] Inoguchi, J., Lee, J.-E.: On the η-parallelism in almost Kenmotsu 3-manifolds. J. Korean Math. Soc., to appear.
  • [66] Inoguchi, J., Lee, S.: A Weierstrass representation for minimal surfaces in Sol. Proc. Amer. Math. Soc. 136, 2209-2216 (2008). https://doi.org/10.1090/S0002-9939-08-09161-2
  • [67] Inoguchi, J., Naitoh, H.: Grassmann geometry on the 3-dimensional non-unimodular Lie groups. Hokkaido Math. J. 48 (2): 385-406 (2019). https://doi.org/10.14492/hokmj/1562810516
  • [68] Janssens, D., Vanhecke, L.: Almost contact structures and curvature tensors. Kodai Math. J. 4 (1), 1-27 (1981). https://doi.org/10.2996/kmj/1138036310
  • [69] Kenmotsu, K.: A class of almost contact Riemannian manifolds. Tohoku Math. J. 24, 93-103 (1972) https://doi.org/10.2748/tmj/1178241594
  • [70] Kenmotsu, K.: Weierstrass formula for surfaces of prescribed mean curvature. Math. Ann. 245, 89-99 (1979). https://doi.org/10.1007/BF01428799
  • [71] Kim, T. W., Pak, H. K.: Canonical foliations of certain classes of almost contact metric structures. Acta Math. Sin. (Engl. Ser.) 21 (4), 841-846 (2005). https://doi.org/10.1007/s10114-004-0520-2
  • [72] Kimura, M., Maeda, S.: On real hypersurfaces of a complex projective space. Math. Z. 202 (3) 299-311 (1989). https://doi.org/10.1007/BF01159962
  • [73] Kiran Kumar, D. L., Nagaraja, H. G., Manjulamma, U., Shashidhar, S.: Study on Kenmotsu manifolds admitting generalized Tanaka-Webster connection. Ital. J. Pure Appl. Math. 47 721-733 (2022).
  • [74] Kokubu, M.: Weierstrass representation for minimal surfaces in hyperbolic space. Tohoku Math. J. 49 (3) 367–377 (1997). https://doi.org/10.2748/tmj/1178225110
  • [75] Kon, M.: Invariant submanifolds in Sasakian manifolds. Math. Ann. 219 (3) 277-290 (1976). https://doi.org/10.1007/BF01354288
  • [76] Kowalski, O.: An explicit classication of 3-dimensional Riemannian spaces satisfying R(X, Y ) · R = 0, Czech. Math. J. 46 (3) 427-474 (1996).
  • [77] Kowalski, O., Sekizawa, M.: Pseudo-symmetric spaces of constant type in dimension three–elliptic spaces. Rend. Mat. VII. 17 477-512 (1997).
  • [78] Milnor, J.: Curvatures of left invariant metrics on Lie groups. Adv. Math. 21 (3) 293-329 (1976). https://doi.org/10.1016/S0001-8708(76)80002-3
  • [79] Nistor, A.I.: Constant angle surfaces in solvable Lie groups. Kyushu J. Math. 68 (2) 315-332 (2014) http://dx.doi.org/10.2206/kyushujm.68.315
  • [80] Okumura, M.: Some remarks on space with a certain contact structure. Tôhoku Math. 14 135-145 (1962). https://doi.org/10.2748/tmj/1178244168
  • [81] Olszak, Z.: On almost cosymplectic manifolds. Kodai Math. J. 4 (2) 239-250 (1981). https://doi.org/10.2996/kmj/1138036371
  • [82] Olszak, Z.: Normal almost contact metric manifolds of dimension three. Ann. Pol. Math. 47 41-50 (1986). https://doi.org/10.4064/ap-47-1-41-50
  • [83] O’Neill, B.: Semi-Riemannian Geometry with Application to Relativity. Academic Press, Orland, 1983.
  • [84] Öztürk, H.: On almost alpha-cosymplectic manifolds with some nullity distributions. Honam Math. J. 41 (2), 269–284 (2019). https://doi.org/10.5831/HMJ.2019.41.2.269
  • [85] Öztürk, H., N. Aktan, N., Murathan, C.: Almost α-cosymplectic (κ, μ, ν)-spaces. Preprint arXiv:1007.0527v1 [math.DG] (2010).
  • [86] Pak, H.-K.: Canonical foliations of almost f-cosymplectic structures, J. Korea Ind. Inf. Syst. Res. 7 (3), 89-94, (2002).
  • [87] Pan, Q.,Wu, H.,Wang, Y.: Almost Kenmotsu 3-h-manifolds with transversely Killing-type Ricci operators. Open Math. 18 (1) 1056-1063 (2020). https://doi.org/10.1515/math-2020-0057
  • [88] Pastore, A. M., Saltarelli, V.: Generalized nullity conditions on almost Kenmotsu manifolds. Int. Electron. J. Geom. 4 168-183 (2011). https://dergipark.org.tr/en/pub/iejg/issue/47488/599509
  • [89] Perrone, D.: Weakly ϕ-symmetric contact metric spaces. Balkan J. Geom. Appl. 7 (2) 67-77 (2002).
  • [90] Perrone, D.: Classification of homogeneous almost cosymplectic three-manifolds. Differ. Geom. Appl. 30 (1) 49-58 (2012). https://doi.org/10.1016/j.difgeo.2011.10.003
  • [91] Perrone, D.: Almost contact metric manifolds whose Reeb vector field is a harmonic section. Acta Math. Hung. 138 (2013), 102-126. https://doi.org/10.1007/s10474-012-0228-1
  • [92] Perrone, D., Remarks on Levi harmonicity of contact semi-Riemannian manifolds. J. Korean Math. Soc. 51 (5), 881-895 (2014). http://dx.doi.org/10.4134/JKMS.2014.51.5.881
  • [93] Perrone, D.: Classification of homogeneous almost α-coKähler three-manifolds. Differ. Geom. Appl. 59 66–90 (2018) 66–90 https://doi.org/10.1016/j.difgeo.2011.10.00
  • [94] Perrone, D.: Left-invariant almost α-coKähler structures on 3D semidirect product Lie groups. Int. J. Geom. Methods Mod. Phys. 16 (1) 1950011 18 pages (2019). https://doi.org/10.1142/S0219887819500117
  • [95] Perrone, D., Almost contact Riemannian three-manifolds with Reeb flow symmetry. Differ. Geom. Appl. 75 (2021), Article ID 101736 11 pages(2021). https://doi.org/10.1016/j.difgeo.2021.101736
  • [96] Rehman, N. A.: Harmonic maps on Kenmotsu manifolds. An. S¸t. Univ. Ovidius Constanta Ser. Math. 21 (3) 197-208 (2013).
  • [97] Saltarelli, V., Three-dimensional almost Kenmotsu manifolds satisfying certain nullity conditions. Bull. Malays. Math. Sci. Soc. 38, 437-459(2015). https://doi.org/10.1007/s40840-014-0029-5
  • [98] Sasaki, S., Hatakeyama, Y.: On differentiable manifolds with certain structures which are close related to almost contact structure II. Tôhoku Math. 13 281-294 (1961). https://doi.org/10.2748/tmj/1178244304
  • [99] Sekigawa, K.: Some 3-dimensional Riemannian manifolds with constant scalar curvature, Sci. Rep. Niigata Univ. Ser. A. 11 25-29 (1974).
  • [100] Szabó, Z. I.: Structure theorems on Riemannian manifolds satisfying R(X, Y ) · R = 0, I. the local version, J. Differ. Geom. 17 531-582 (1982).
  • [101] Tachibana, S.: A theorem on Riemannian manifolds of positive curvature operator. Proc. Japan Acad. 50, 301-302 (1974). https://doi.org/10.3792/pja/1195518988
  • [102] Takahashi, T.: Sasakian ϕ-symmetric spaces. Tohoku Math. J. (2) 29 (1) 91-113 (1977). https://doi.org/10.2748/tmj/1178240699
  • [103] Tanno, S.: The automorphism groups of almost contact Riemannian manifolds. Tohoku Math. J. (2) 21 (1), 21-38 (1969). https://doi.org/10.2748/tmj/1178243031
  • [104] Tanno, S.: Variational problems on contact Riemannian manifolds. Trans. Amer. Math. Soc. 314 349-379 (1989). https://doi.org/10.1090/S0002-9947-1989-1000553-9
  • [105] Tasaki, H., Umehara, M.: An invariant on 3-dimensional Lie algebras. Proc. Amer. Math. Soc. 115 (2) 293-294 (1992). https://doi.org/10.1090/S0002-9939-1992-1087471-0
  • [106] Tricerri, F. and Vanhecke, L., Homogeneous Structures on Riemannian Manifolds, Lecture Notes Series, London Math. Soc. 52, (1983), Cambridge Univ. Press.
  • [107] Vaisman, I.: Conformal changes of almost contact metric structures, in: R. Artzy, I. Vaisman (Eds.), Geometry and Differential Geometry (Artzy, R., Vaisman, I., Eds.), Lecture Notes in Math. 792, Springer Verlag, 435–44, (1980).
  • [108] Voicu, R. C.: Ricci curvature properties and stability on 3-dimensional Kenmotsu manifolds. Harmonic maps and differential geometry, 273- 278, Contemp. Math., 542, Amer. Math. Soc., Providence, RI, 2011.
  • [109] Wang, Y.: Three-dimensional locally symmetric almost Kenmotsu manifolds, Ann. Pol. Math. 116 (1), 79-86 (2016). https://doi.org/10.4064/ap3555-12-2015
  • [110] Wang, Y.: A class of 3-dimensional almost Kenmotsu manifolds with harmonic curvature tensors. Open Math. 14, 977-985 (2016).
  • [111] Wang, Y., Liu, X.: Locally symmetric CR-integrable almost Kenmotsu manifolds, Mediterr. J. Math. 12 159-171 (2015). https://doi.org/10.1007/s00009-014-0388-z
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Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Jun-ichi Inoguchı 0000-0002-6584-5739

Early Pub Date October 6, 2023
Publication Date October 29, 2023
Acceptance Date September 22, 2023
Published in Issue Year 2023

Cite

APA Inoguchı, J.-i. (2023). Characteristic Jacobi Operator on Almost Kenmotsu $3$-manifolds. International Electronic Journal of Geometry, 16(2), 464-525. https://doi.org/10.36890/iejg.1300339
AMA Inoguchı Ji. Characteristic Jacobi Operator on Almost Kenmotsu $3$-manifolds. Int. Electron. J. Geom. October 2023;16(2):464-525. doi:10.36890/iejg.1300339
Chicago Inoguchı, Jun-ichi. “Characteristic Jacobi Operator on Almost Kenmotsu $3$-Manifolds”. International Electronic Journal of Geometry 16, no. 2 (October 2023): 464-525. https://doi.org/10.36890/iejg.1300339.
EndNote Inoguchı J-i (October 1, 2023) Characteristic Jacobi Operator on Almost Kenmotsu $3$-manifolds. International Electronic Journal of Geometry 16 2 464–525.
IEEE J.-i. Inoguchı, “Characteristic Jacobi Operator on Almost Kenmotsu $3$-manifolds”, Int. Electron. J. Geom., vol. 16, no. 2, pp. 464–525, 2023, doi: 10.36890/iejg.1300339.
ISNAD Inoguchı, Jun-ichi. “Characteristic Jacobi Operator on Almost Kenmotsu $3$-Manifolds”. International Electronic Journal of Geometry 16/2 (October 2023), 464-525. https://doi.org/10.36890/iejg.1300339.
JAMA Inoguchı J-i. Characteristic Jacobi Operator on Almost Kenmotsu $3$-manifolds. Int. Electron. J. Geom. 2023;16:464–525.
MLA Inoguchı, Jun-ichi. “Characteristic Jacobi Operator on Almost Kenmotsu $3$-Manifolds”. International Electronic Journal of Geometry, vol. 16, no. 2, 2023, pp. 464-25, doi:10.36890/iejg.1300339.
Vancouver Inoguchı J-i. Characteristic Jacobi Operator on Almost Kenmotsu $3$-manifolds. Int. Electron. J. Geom. 2023;16(2):464-525.