Homogeneous Riemannian Structures in Thurston Geometries and Contact Riemannian Geometries

Year 2024, Volume: 17 Issue: 2, 559 - 659

Jun-ichi Inoguchi

https://doi.org/10.36890/iejg.1464086

Abstract

We give explicit parametrizations for all the homogeneous Riemannian structures on model spaces of Thurston geometry. As an application, we give all the homogeneous contact metric structures on $3$-dimensional Sasakian space forms.

Keywords

Homogeneous structures, Thurston geometry, Sasakian manifolds, Ambrose-Singer connections

References

• [1] Abe, K.: The classification of homogeneous structures on 3-dimensional space forms, Math. J. Okayama Univ. 28 (1), 173-189 (1986).
• [2] Agricola, T, Friedrich, F.: On the holonomy of connections with skew-symmetric torsion, Math. Ann. 328,711-748 (2004).
• [3] Alekseevsky, D., Gorodski, C.: Semisimple symmetric contact spaces, Indag. Math. New Ser. 31 (6), 1110-1133 (2020).
• [4] Ambrose, W., Palais, R.S., Singer, I. M.: Sprays, Ann. Acad. Brasil. Ciˇencias 32, 163-178 (1960).
• [5] Ambrose, W., Singer, I. M.: On homogeneous Riemannian manifolds, Duke Math. J. 25, 647-669 (1958).
• [6] Auslander, L., Green, L., Hahn, F.: Flows on homogeneous spaces, with the assistance of L. Markus and W. Massey, and an appendix by L. Greenberg, Ann. of Math. Stud. 53 Princeton University Press, Princeton, NJ, 1963.
• [7] Azencott, R., Wilson, E. N.: Homogeneous manifolds with negative curvature. I, Trans. Am. Math. Soc. 215, 323-362 (1976).
• [8] Azencott, R., Wilson, E. N.: Homogeneous manifolds with negative curvature. II, Mem. Am. Math. Soc. 178 (1976).
• [9] Bazdar, A., Teleman, A.: Infinitesimal homogeneity and bundles, Ann Glob Anal Geom 59, 197-231 (2021).
• [10] Belkhelfa, M., Dillen, F., Inoguchi, J.: Surfaces with parallel second fundamental form in Bianchi-Cartan-Vranceanu spaces, PDE’s, Submanifolds and Affine Differential Geometry (Warsaw, 2000), Banach Center Publ., 57, 67-87, (2002).
• [11] Bérad-Bergery, L., Bourguignon, J. P.: Laplacians and Riemannian submersions with totally geodesic fibers, Illinois J. Math. 26, 181-200 (1982).
• [12] Berger, M.: Les Varietes Riemanniennes homogenes normlales simplement connexes a courbure strictement positive, Annli Scuola Norm. Sup. Pisa 15, 179-246 (1961).
• [13] Bianchi, L.: Sugli sazi a tre dimensioni che ammettono un gruppo continuo di movimenti, Memorie di Matematica e di Fisica della Societa Italiana delle Scienze, Serie Tereza, Tomo XI, 267-352 (1898); English translation: On the three-dimensional spacse which admit a continuous group of motions, General Relativity and Gravitation 33 (12), 2171-2252 (2001).
• [14] Bieliavsky, P., Falbel, E., Gorodski, C.: The classification of simply-connected contact sub-Riemannian symmetric spaces, Pacific J. Math. 188 (1), 65-82 (1999).
• [15] Blair, D. E.: Almost contact manifolds with Killing structure tensors, Pacific J. Math. 39 (2), 285-292 (1971).
• [16] Blair, D. E.: Contact Manifolds in Riemannian Geometry, Lecture Notes in Math. 509, Springer-Verlag, Berlin-Heidelberg-New-York, (1976).
• [17] Blair, D. E.: Riemannian Geometry of Contact and Symplectic Manifolds, Progress in Math. 203, Birkhäuser, Boston, Basel, Berlin, (2002).
• [18] Blair, D.E., Koufogiorgos, T., Papantoniou, B.J.: Contact metric manifolds satisfying a nullity condition, Israel J. Math. 91, 189-214 (1995).
• [19] Blair, D.E., Showers, D.K.: Almost contact manifolds with Killing structure tensors. II, J. Differ. Geom. 9, 577-582 (1974).
• [20] Blair, D. E., Vanhecke, L.: Symmetries and φ-symmetric spaces, Tôhoku Math. J. (2) 39 (3), 373-383 (1987).
• [21] Blair, D. E., Vanhecke, L.: New characterizations of φ-symmetric spaces, Kodai Math. J. 10 (1), 102-107 (1987).
• [22] Blair, D. E., Vanhecke, L.: Volume-preserving φ-geodesic symmetries, C. R. Math. Acad. Sci. Soc. R. Can. 9, 31-36 (1987).
• [23] Boeckx, E.: A class of locally φ-symmetric contact metric spaces, Arch. Math. 72 (6), 466-472 (1999).
• [24] Boeckx, E.: A full classification of contact metric (κ, μ)-spaces, Illinois J. Math. 44, 212-219 (2000).
• [25] Boeckx, E., Bueken, P., Vanhecke, L.: Flow-symmetric Riemannian manifolds, Beitr. Algebra Geom. 40, (2), 459-474 (1999).
• [26] Boeckx, E., Cho, J. T.: Locally symmetric contact metric manifolds, Monatsh. Math. 148 (4), 269-281 (2006).
• [27] Boeckx, E., Cho, J. T.: Pseudo-Hermitian symmetries, Israel J. Math. 166, 125-145 (2008).
• [28] Boothby, W. M., Wang, H. C.: On contact manifolds, Ann. Math. (2) 68, 721-734 (1958).
• [29] Boyer, C. P., Galicki, K.: Sasakian Geometry, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2008.
• [30] Bueken, P., Vanhecke, L.: Geometry and symmetry on Sasakian manifolds, Tsukuba J. Math. 12 (2), 403-422 (1988).
• [31] Bueken, P., Vanhecke, L.: Harmonic reflections on Sasakian manifolds, Math. J. Okayama Univ. 30, 187-197 (1988).
• [32] Bueken, P., Vanhecke, L.: Reflections in K-contact geometry, Math. Rep. Toyama Univ. 12, 41-49 (1989).
• [33] Bueken, P., Vanhecke, L.: Rotations and harmonicity in contact geometry, Rend. Mat. Appl., VII. 12 (1), 127-141 (1992).
• [34] Burstall, F. E., Rawnsley, J. H.: Twistor Theory for Riemannian Symmetric Spaces With Applications to Harmonic Maps of Riemann Surfaces, Lecture Notes in Math. 1424, Springer Verlag, (1990).
• [35] Cabrerizo, J. L. Fernández, M., Gómez, J. S.: On the existence of almost contact structure and the contact magnetic field, Acta Math. Hung. 125 (1-2), 191-199 (2009).
• [36] Calvaruso, C., Castrillón López, M.: Pseudo-Riemannian Homogeneous Structures, Springer Verlag, (2019).
• [37] Calvaruso, G., Fino, A.: Five-dimensional K-contact Lie algebras, Monatsh. Math. 167 (1), 35-59 (2012).
• [38] Cartan, E.: Leçon sur la geometrie des espaces de Riemann, Second Edition, Gauthier-Villards, Paris, (1946).
• [39] Calviño-Louzao, E., Ferreiro-Subrido, M., García-Río, E., Vázquez-Lorenzo, R.: Homogeneous Riemannian structures in dimension three, Rev. Real. Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 117, Article number 70 (2023).
• [40] Castrillón López, M., M. Gadea, P. M., Swann, A.: Homogeneous structures on real and complex hyperbolic spaces, Illinois J. Math. 53 (2), 561-574 (2009).
• [41] Castrillón López, M., M. Gadea, P. M., Swann, A.: The homogeneous geometries of real hyperbolic space, Mediter. J. Math. 10 (2), 1011-1022 (2013).
• [42] Castrillón López, M., Luján, I.: Reduction of homogeneous Riemannian structures, Proc. Edinb. Math. Soc. (2) 58 (1), 81-106 (2015).
• [43] Chen, B. Y., Vanhecke, L.: Isometric, holomorphic and symplectic reflections, Geom. Dedicata 29 (3), 259-277 (1989).
• [44] Chinea, D., Gonzalez, C.: An example of an almost cosymplectic homogeneous manifolds, Differential Geometry Peñiscola 1985, Lecture Notes in Math. 1209, 133-142 (1986).
• [45] Chinea, D., Gonzalez, C.: A classification of almost contact metric manifolds, Ann. Mat. Pura Appl. (4) 156, 15-36 (1990).
• [46] Chinea, D., Gonzalez, C.: Classification of almost contact metric structures, Rev. Roum. Math. Pures Appl. 37 (3), 199–211 (1992).
• [47] Cho, J. T.: On some classes of almost contact metric manifolds, Tsukuba J. Math. 19 (1), 201-217 (1995).
• [48] Cho, J. T., Inoguchi, J.: Pseudo-symmetric contact 3-manifolds, J. Korean Math. Soc. 42 (5), 913-932 (2005).
• [49] Cho, J. T., Inoguchi, J.: Pseudo-symmetric contact 3-manifolds. II. When is the tangent sphere bundle over a surface pseudo-symmetric ?, Note Mat. 27 (1), 119-129 (2007).
• [50] Cho, J. T., Inoguchi, J.: Curvatures and symmetries of tangent sphere bundles, Houston J. Math. 37 (4), 1125-1142.
• [51] Cho, J. T., Inoguchi, J., Lee, J.-E.: Pseudo-symmetric contact 3-manifolds. III, Colloq. Math. 114 (1), 77–98 (2009).
• [52] Cleyton, R., Moroianu, A., Semmelmann, U.: Metric connections with parallel skew-symmetric torsion, Adv. Math. 378, 107519 (2021).
• [53] D’Atri, J. E., Nickerson, H. K.: Geodesic symmetries in spaces with special curvature tensor, J. Diff. Geom. 9, 251-262 (1974).
• [54] D’Atri, J. E., Nickerson, H. K.: Geodesic spheres and symmetries in naturally reductive homogeneous spaces, Michigan Math. J. 22, 71-76 (1975).
• [55] D’Atri, J. E., Ziller, W.: Naturally Reductive Metrics and Einstein Metrics on Compact Lie Groups, Mem. Am. Math. Soc. 215 (1979).
• [56] De Nicola, A., Dileo, G., Yudin, I.: On nearly Sasakian and nearly cosymplectic manifolds, Ann. Mat. Pura Appl. (4) 197 (1), 127-138 (2018).
• [57] Dileo, G., Lotta, A.: A classification of spherical symmetric CR manifolds, Bull. Aust. Math. Soc. 80, 251-274 (2009).
• [58] Dombrowski, P.: On the geometry of the tangent bundle, J. Reine Angew. Math. 210, 73-88 (1962).
• [59] Dorfmeister, J. F., Inoguchi, J., Kobayashi, S.-P.: Constant mean curvature surfaces in hyperbolic 3-space via loop groups, J. Reine Angew. Math. 686 (1), 1-36 (2014).
• [60] Dorfmeister, J. F., Inoguchi, J., Kobayashi, S.-P.: A loop group method for affine harmonic maps into Lie groups, Adv. Math. 298, 207-253 (2016).
• [61] Dorfmeister, J. F., Inoguchi, J., Kobayashi, S.-P.: A loop group method for minimal surfaces in the three-dimensional Heisenberg group, Asian J. Math. 20 (3), 409-448 (2016).
• [62] Draper, C., Garvín, A., Palomo, F. J.: Invariant affine connections on odd-dimensional spheres, Ann. Glob. Anal. Geom. 49, 213-251 (2016).
• [63] Eliashberg, Ya.: Contact 3-manifolds twenty years since J. Martinet’s work, Ann. l’institut Fourier 42 (1-2), 165-192 (1992).
• [64] Erjavec, Z., Inoguchi, J.: Geodesics and magnetic curves in the 4-dim almost Kähler model space F4, Complex Manifolds 11, Article ID 20240001, 33 p. (2024).
• [65] Falbel, E., Gorodski, C.: On contact sub-Riemannian symmetric spaces, Ann. Sci. Éc.Norm. Supér. (4) 28 (5), 571–589 (1995).
• [66] Falbel, E., Gorodski, C.: Sub-Riemannian homogeneous spaces in dimensions 3 and 4, Geom. Dedicata 62 (3), 227-252 (1996).
• [67] Falbel, E., Gorodski, C., Veloso, J. M.: Conformal sub-Riemannian geometry in dimension 3, Mat. Contemp. 9, 61-73 (1995).
• [68] Ferus, D., Pinkall, U.: Constant curvature 2-spheres in the 4-sphere, Math. Z. 200 (2), 265-271 (1989).
• [69] Fino, A.: Almost contact homogeneous manifolds, Riv. Mat. Univ. Parma (5) 3, 321-332 (1994).
• [70] Fino, A.: Almost contact homogeneous structures, Boll. Un. Mat. Ital. A (7) 9, 299–311 (1995).
• [71] Fino, A., Grantcharov, G.: Properties of manifolds with skew-symmetric torsion and special holonomy, Adv. Math. 189 (2), 439-450 (2004).
• [72] Foreman, B. J.: K-contact Lie groups of dimension five or greater, Kodai Math. J. 34 (1), 79-84 (2011).
• [73] Friedrich, T., Ivanov, S.: Parallel spinors and connections with skew-symmetric torsion in string theory, Asian J. Math. 6 (2), 303-335 (2002).
• [74] Friedrich, T., Ivanov, S.: Almost contact manifolds, connections with torsion, and parallel spinors, J. Reine Angew. Math. 559, 217-236 (2003).
• [75] Fujioka, A., Inoguchi, J.: Spacelike surfaces with harmonic inverse mean curvature, J. Math. Sci. Univ. Tokyo 7, 657-698 (2000).
• [76] Gadea, P. M., González-Dávila, J. C., Oubiña, J. A.: Cyclic metric Lie groups, Monatsh. Math. 176 (2), 219-239 (2015).
• [77] Gadea, P. M., González-Dávila, J. C., Oubiña, J. A.: Cyclic homogeneous Riemannian manifolds, Ann. Mat. Pura Appl. (4) 195 (5), 1619-1637 (2016).
• [78] Gadea, P. M., González-Dávila, J. C., Oubiña, J. A.: Homogeneous spin Riemannian manifolds with the simplest Dirac operator, Adv. Geom. 18 (3), 289-302 (2018).
• [79] Gadea, P. M., Oubiña, J. A.: Homogeneous Riemannian structures on Berger 3-spheres, Proc. Edinb. Math. Soc. (2) 48, (2), 375-387 (2005).
• [80] Gadea, P. M., Oubiña, J. A.: Homogeneous Kähler and Sasakian structures related to complex hyperbolic spaces, Proc. Edinb. Math. Soc. (2) 53 (2), 393-413 (2010).
• [81] Geiges, G.: An Introduction to Contact Topology, Cambridge Studies in Advanced Mathematics 109, (2008).
• [82] Gray, A.: Riemannian manifolds with geodesic symmetries of order 3, J. Differ. Geom. 7, 343-369 (1972).
• [83] Gonzalez, C., Chinea, D.: Quasi-Sasakian homogeneous structures on the generalized Heisenberg group H(p, 1), Proc. Am. Math. Soc. 105, 173-184 (1989).
• [84] Gordeeva, I.A., Pan’zhenskii, V.I., Stepanov, S.E.: Riemann–Cartan manifolds. J. Math. Sci. 169, 342-361 (2010).
• [85] Gordon, C. S.: Naturally reductive homogeneous Riemannian manifolds, Canadian J. Math. 37 (3), 467-487 (1985).
• [86] Ha, K.Y., Lee, J.B.: Left invariant metrics and curvatures on simply connected three-dimensional Lie groups, Math. Nachr. 282 (6), 868-898 (2009).
• [87] Halverscheid, S., Iannuzzi, A.: On naturally reductive left-invariant metrics of SL(2,R), Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 5 (2), 171-187 (2006).
• [88] Hasegawa, Y.: C-flows on a Lie group for Euler equations, Nagoya Math. J. 40, 67-84 (1970).
• [89] Hassani M., Ahmadi, P.: Isometric actions on the four dimensional Minkowski spacetime, Differential Geom. Appl. 88, Article number 102007, (2023).
• [90] Hatakeyama, Y.: Some notes on differentiable manifolds with almost contact structures, Tôhoku Math. J. (2) 15, 176-181 (1963).
• [91] Heintze, E.: On homogeneous manifolds of negative curvature, Math. Ann. 211, 23-34 (1974).
• [92] Helgason, S.: Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics 34, American Mathematical Society (2001).
• [93] Hsu, C.: On some structures which are similar to the quaternion structure, Tôhoku Math. J. (2) 12, 403–428 (1960).
• [94] Inoguchi, J.: Timelike surfaces of constant mean curvature in Minkowski 3-space, Tokyo J. Math. 21, 141-152 (1998).
• [95] Inoguchi, J.: Minimal surfaces in 3-dimensional solvable Lie groups, Chinese Ann. Math. B. 24, 73-84 (2003).
• [96] Inoguchi, J.: Invariant minimal surfaces in real special linear group of degree 2, Ital. J. Pure Appl. 16, 61-80 (2004).
• [97] Inoguchi, J.: Minimal surfaces in 3-dimensional solvable Lie groups II, Bull. Austral. Math. Soc. 73, 365-374 (2006).
• [98] 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
• [99] Inoguchi, J.: On homogeneous contact 3-manifolds, Bull. Fac. Edu. Utsunomiya Univ. Sect. 2 59, 1-12 (2009). http://hdl.handle.net/10241/00004788
• [100] Inoguchi, J.: A note on almost contact Riemannian 3-manifolds, Bull. Yamagata Univ. Nat. Sci. 17 (1), 1-6 (2010).
• [101] Inoguchi, J.: A note on almost contact Riemannian 3-manifolds. II, Bull. Korean Math. Soc. 54 (1), 85-97 (2017).
• [102] Inoguchi, J.: Characteristic Jacobi operator on almost cosymplectic 3-manifolds, Internat. Elect. J. Geom. 12 (2), 276-299 (2019).
• [103] Inoguchi, J.: Characteristic Jacobi operator on almost Kenmotsu 3-manifolds, Internat. Elect. J. Geom. 16 (2), 464–525 (2023).
• [104] Inoguchi, J.: Differential geometry of the unit tangent sphere bundle over the 3-sphere, in preparation.
• [105] Inoguchi, J., Kumamoto, T., Ohsugi, N., Suyama, Y.: Differential Geometry of curves and surfaces in 3-dimensional homogeneous spaces I, Fukuoka Univ. Sci. Rep., 29 (2), 155-182 (1999).
• [106] Inoguchi, J., Kumamoto, T., Ohsugi, N., Suyama, Y.: Differential Geometry of curves and surfaces in 3-dimensional homogeneous spaces II, Fukuoka Univ. Sci. Rep., 30 (1), 17-48 (2000).
• [107] Inoguchi, J., Kumamoto, T., Ohsugi, N., Suyama, Y.: Differential Geometry of curves and surfaces in 3-dimensional homogeneous spaces III, Fukuoka Univ. Sci. Rep., 30 (20), 130-160 (2000).
• [108] Inoguchi, J., Kumamoto, T., Ohsugi, N., Suyama, Y.: Differential Geometry of curves and surfaces in 3-dimensional homogeneous spaces IV, Fukuoka Univ. Sci. Rep., 30 (20), 161-168 (2000).
• [109] Inoguchi, J., Kuwabara, K., Naitoh, H.: Grassmann geometry on the 3-dimensional Heisenberg group, Hokkaido Math. J. 34 (2), 375-391 (2005).
• [110] Inoguchi, J., Lee, J,-E.: Affine biharmonic curves in 3-dimensional homogeneous geometries, Mediterr. J. Math. 10 (2013), 571-592 (2013).
• [111] Inoguchi, J., Lee, S.: A Weierstrass representation for minimal surfaces in Sol., Proc. Am. Math. Soc. 136, 2209-2216 (2008).
• [112] Inoguchi, J., Munteanu, M. I.: Periodic magnetic curves in Berger spheres, Tôhoku Math. J. (2) 69 (1), 113–128 (2017).
• [113] Inoguchi, J., Munteanu, M. I.: Magnetic curves in tangent sphere bundles. II, J. Math. Anal. Appl. 466 (2), 1570-1581.
• [114] Inoguchi, J., Munteanu, M. I.: Magnetic curves in the real special linear group, Adv. Theor. Math. Phys. 23 (8), 2161-2205 (2019).
• [115] Inoguchi, J., Munteanu, M. I.: Magnetic curves in tangent sphere bundles, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 113 (3), 2087-2112 (2019).
• [116] Inoguchi, J., Munteanu, M. I.: Homogeneity of magnetic trajectories in the real special linear group, Proc. Am. Math. Soc. 152 (3), 1287–1300, (2024).
• [117] Inoguchi, J., Munteanu, M. I.: Magnetic curves in tangent sphere bundles. III, submitted.
• [118] Inoguchi, J., Munteanu, M. I.: Homogeneity of magnetic trajectories in the Berger sphere, submitted.
• [119] Inoguchi, J., Munteanu, M. I.: The oscillator group and homogeneous magnetic trajectories in the Heisenberg group, submitted.
• [120] Inoguchi, J., Naitoh, H.: Grassmann geometry on the 3-dimensional unimodular Lie groups I, Hokkaido Math. J. 38 (3), 427-496 (2009).
• [121] Inoguchi, J., Naitoh, H.: Grassmann geometry on the 3-dimensional unimodular Lie groups II, Hokkaido Math. J. 40 (3), 411-429 (2011).
• [122] Inoguchi, J., Naitoh, H.: Grassmann geometry on the 3-dimensional non-unimodular Lie groups, Hokkaido Math. J. 48 (2), 385-406 (2019).
• [123] Inoguchi, J., Naitoh, H.: Grassmann geometry on H2 × R, submitted.
• [124] Inoguchi, J., Ohno, Y.: Homogeneous structures of 3-dimensional Sasakian space forms, Tsukuba J. Math., to appear.
• [125] Inoguchi, J., Ohno, Y.: Homogeneous structures of 3-dimensional Lie groups, submitted.
• [126] Inoguti, J.: Rotations and φ-symmteric spaces, Math. J. Toyama Univ. 15, 123-130 (1992).
• [127] Inoguti, J.: Nearly cosymplectic manifolds with rotations of order 3, preprint, (1993).
• [128] Inoguti, J., Sekizawa, M.: Symmetries which preserve the characteristic vector fields of K-contact manifolds, Note Mat. 13 (2), 229-236 (1993).
• [129] Ise, M., Takeuchi, M.: Lie groups I. Lie groups II. (Translated by Nomizu), Translations of Mathematical Monographs. 85, American Mathematical Society (1991).
• [130] Itoh, M.: Invariant connections and Yang–Mills solutions, Trans. Am. Math. Soc. 267 (1981), 229-236 (1981).
• [131] Janssense, D., Vanhecke, L.: Almost contact structures and curvature tensors, Kodai Math. J. 4, 1-27 (1981).
• [132] Jiménez, J. A.: Existence of Hermitian n-symmetric spaces of non-commutative naturally reductive spaces, Math. Z. 196, 133-139 (1987). Addendum: Math. Z. 197, (3), 455-456 (1988).
• [133] Jiménez, J. A.: The focal locus of a Riemannian 4-symmetric space, Can. Math. Bull. 31 (2), 175-181 (1988).
• [134] Jiménez, J. A.: Riemannian 4-symmetric spaces, Trans. Am. Math. Soc. 306 (2), 715-734 (1988).
• [135] Jiménez, J. A., Kowalski, O.: The classification of φ-symmetric Sasakian manifolds, Monatsh. Math. 115 (1-2), 83-98 (1993).
• [136] Kato, T., Motomiya, K.: A study on certain homogeneous spaces, Tohôku Math. J. (2) 21, 1-20 (1969).
• [137] Katsuda, A.: A pinching problem for locally homogeneous spaces, J. Math. Soc. Japan 41 (1), 57-74 (1989).
• [138] Kaup, W., Zaitsev, D.: On symmetric Cauchy–Riemann manifolds, Adv. Math. 149, 145-181 (2000).
• [139] Kenmotsu, K.: A class of almost contact Riemannian manifolds, Tôhoku Math. J. (2) 24 (1), 93-103 (1972).
• [140] Kim, I. B., Takahashi, T.: Isoparametric hypersurfaces in a space form and metric connections, Tsukuba J. Math. 21 (1), 15-28 (1997).
• [141] Klingenberg, W., Sasaki, S.: The tangent sphere bundle of a 2-sphere, Tôhoku Math. J. 27 (1), 45-57 (1975).
• [142] Kriˇchenko, V. F.: On homogeneous Riemannian spaces with invariant tensor structures, Sov. Math. Dokl. 21, 734-737 (1980).
• [143] Kobayashi, S.: Principal fibre bundles with the 1-dimensional toroidal group, Tôhoku Math. J. (2) 8, 29-45 (1956).
• [144] Kobayashi, S.: Homogeneous Riemannian manifolds of negative curvature, Tôhoku Math. J. (2) 14, 413-415 (1962).
• [145] Kobayashi, S., Nomizu, K.: Foundations of differential geometry. II, Interscience Tracts in Pure and Applied Mathematics 15, New York- London: Interscience Publishers, a division of John Wiley and Sons. (1969).
• [146] Koda, T.,Watanabe, Y.: Homogeneous almost contact Riemannian manifolds and infinitesimal models, Boll. Un. Mat. Ital. (7) 11-B, suppl., 11-24 (1997).
• [147] Kodama, H., Takahara, A., Tamaru, H.: The space of left-invariant metrics on a Lie group up to isometry and scaling, Manuscr. Math. 135 (1-2), 229-243 (2011).
• [148] Kostant, B.: A characterization of invariant affine connections, Nagoya Math. J. 16, 35-50 (1960).
• [149] Kowalski, O.: Generalized Symmetric Spaces, Lecture Notes in Math. 805, Springer Verlag (1980).
• [150] Kowalski, O.: On strictly locally homogeneous Riemannian manifolds, Differential Geom. Appl. 7 (2), 131-137 (1997).
• [151] Kowalski, O., Szenthe, J.: On the existence of homogeneous geodesics in homogeneous Riemannian manifolds, Geom. Dedicata 81, 209-214 (2000).
• [152] Kowalski, O., Tricerri, F.: Riemannian manifolds of dimension n ≤ 4 admitting a homogeneous structure of class T2, Conf. Semin. Mat. Univ. Bari 222, 24 pages. (1987).
• [153] Kowalski, O., Vanhecke, L.: A generalization of a theorem on naturally reductive homogeneous spaces, Proc. Am. Math. Soc. 91(3), 433-435 (1984).
• [154] Kowalski, O., Vanhecke, L.: Riemannian manifolds with homogeneous geodesics, Boll. Unione Mat. Ital., VII. Ser., B 5 (1), 189-246 (1991).
• [155] Kowalski, O., W˛egrzynowski, S.: A classification of five-dimensional ϕ-symmetric spaces, Tensor (N.S.) 46, 379-386 (1987).
• [156] Kurihara, H., Tojo, K.: Involutions of compact Riemannian 4-symmetric spaces, Osaka J. Math. 45 (3), 643-689 (2008).
• [157] Kurihara, H., Tojo, K.: Involutions on a compact 4-symmetric space of exceptional type, Osaka J. Math. 552 (4), 1101-1125 (2015).
• [158] Kuwabara, K.: Grassmann geometry on the groups of rigid motions on the Euclidean and the Minkowski planes, Tsukuba J. Math. 30 (1), 49-59 (2006).
• [159] Maeda, S., Tanabe, H.: Redefinition of Berger spheres from the viewpoint of submanifold geometry, Tôhoku Math. J., to appear.
• [160] Martín Cabrera, F.: On the classification of almost contact metric manifolds, Differ. Geom. Appl. 64, 13-28 (2019).
• [161] Matsuzoe, H., Inoguchi, J.: Statistical structures on tangent bundles, Appl. Sci. 5 (1), 55-57 (2003).
• [162] Meeks, W.H., III. and Pérez, J., Constant mean curvature surfaces in metric Lie groups, Geometric Analysis: Partial Differential Equations and Surfaces, Contemp. Math. 570, 25-110 (2012).
• [163] Milnor, J.: Curvatures of left invariant metrics on Lie groups, Adv. Math. 21, 293-329 (1976).
• [164] Morimoto, A.: On normal almost contact structures, J. Math. Soc. Japan 15, 420-436 (1963).
• [165] Motomiya, K.: A study on almost contact manifolds, Tôhoku Math. J. (2) 20, 73-90 (1968).
• [166] Ni, L., Zheng, F.: On Hermitian manifolds whose Chern connection is Ambrose-Singer, Trans. Am. Math. Soc. 376 (9), 6681-6707 (2023).
• [167] Nicolodi, L.; Vanhecke, L.: Rotations and Hermitian symmetric spaces, Monatsh. Math. 109 (4), 279-291 (1990).
• [168] Nicolodi, L.; Vanhecke, L.: Rotations on a Riemannian manifold, Recent topics in differential geometry, Proc. Workshop/Puerto de la Cruz/Spain 1990, Ser. Inf. 32, 89-101 (1991).
• [169] Nicolodi, L.; Vanhecke, L.: Harmonic and isometric rotations around a curve, Illinois J. Math. 37 (1), 85-100.
• [170] Nomizu, K.: Invariant affine connections on homogeneous spaces, Am. J. Math. 76, 33-56 (1954).
• [171] Ogiue, K.: On fiberings of almost contact manifolds, Kodai Math. Sem. Rep. 17 (1), 53-62 (1965).
• [172] Ohnita, Y.: Canonical connections of a Sasakian manifold and invariant submanifolds with parallel second fundamental form, Proceedings of The 23rd International Differential Geometry Workshop on Submanifolds in Homogeneous Spaces and Related Topics 23, 31-40 (2021).
• [173] Ohnita, Y.: Parallel Kähler submanifolds and R-spaces, Differential Geometry and Global Analysis in Honor of Tadashi Nagano, Contemp. Math. 777, 163-184 (2022).
• [174] Ohno, Y.: Homogeneous structures on S2 × R and H2 × R, Tsukuba J. Math. 47 (2) 239-246 (2023).
• [175] Okumura, M.: Some remarks on space with a certain contact structures, Tôhoku Math. J. (2) 14, 135-145 (1962).
• [176] Olmos, C., Reggiani, S.: The skew-torsion holonomy theorem and naturally reductive spaces, J. Reine Angew. Math. 664, 29-53 (2012).
• [177] Olmos, C., Reggiani, S.: A note on uniqueness of the canonical connection of a naturally reductive space, Monats. Math. 172 (3-4), 379–386 (2013).
• [178] Olmos, C., Sánchez, C.: A geometric characterization of the orbits of s-representations, J. Reine Angew. Math. 420, 195-202 (1991).
• [179] Olszak, Z.: Normal almost contact manifolds of dimension three, Ann. Pol. Math. 47, 42-50 (1986).
• [180] Omori, H.: Infinite dimensional Lie Transformation Groups, Lecture Notes in Math. 427, Springer-Verlag, Berlin, (1974).
• [181] Omori, H.: Infinite-dimensional Lie Groups, Translations of Mathematical Monographs 158, American Mathematical Society, Providence, RI, 1997.
• [182] O’Neill, B.: Semi Riemannian Geometry with Application to Relativity, Academic Press, (1983).
• [183] Pastore, A. M.: On the homogeneous Riemannian structures of type T1 ⊕ T3, Geom. Dedicata 30, 235-246 (1989).
• [184] Pastore, A. M.: Reducibility of homogeneous Riemannian structures of the class T3 in low dimension, Geom. Dedicata 38 (2), 121-136 (1991).
• [185] Pastore, A. M.: Canonical connections with an algebraic curvature tensor field on naturally reductive spaces, Geom. Dedicata 43 (3), 351-361 (1992).
• [186] Pastore, A. M.: Homogeneous representations of the hyperbolic spaces related to homogeneous structures of class T1 ⊕ T3, Rend. Mat. Appl. (7) 12, (2), 445–453 (1992).
• [187] Pastore, A. M., Verroca, F.: Some results on the homogeneous Riemannian structures of class T1 ⊕ T2, Rend. Mat. Appl. (7) 11 (1), 105-121 (1991).
• [188] Patrangenaru, V.: Classifying 3- and 4-dimensional homogeneous Riemannian manifolds by Cartan triples, Pacific J. Math. 173, 511-532 (1996).
• [189] Patrangenaru, V.: 5 dimensional strictly locally homogeneous Riemannian manifolds Period. Math. Hungar. 45 (1-2), 123-129(2002).
• [190] Perrone, D.: Homogeneous contact Riemannian three-manifolds, Illinois J. Math. 42, 243-256 (1998).
• [191] Perrone, D.: Classification of homogeneous almost cosymplectic three-manifolds, Differ. Geom. Appl. 30 (1), 49-58 (2012).
• [192] Perrone, D.: Classification of homogeneous almost α-coKähler three-manifolds, Differ. Geom. Appl. 59, 66-90 (2018).
• [193] Perrone, D.: Left-invariant almost α-coKähler structures on 3D semidirect product Lie groups, Int. J. Geom. Methods Mod. Phys. 16 (1), Article ID 1950011, 18 pp. (2019).
• [194] Prüfer, F.: On compact Riemannian manifolds with volume-preserving symmetries, Ann. Glob. Anal. Geom. 7, 133-140 (1989).
• [195] Rastrepina, A. O., Surina, O. P.: Invariant almost contact structures and connections on the Lobachevsky space (Russian), Izv. Vyssh. Uchebn. Zaved. Mat. (2023), (2), 47–56 (2023). English translation: Russian Math. (Iz. VUZ) 67 (2), 43-51 (2023).
• [196] Salvai, M.: Spectra of unit tangent bundles of compact hyperbolic Riemann surfaces, Ann. Global Anal. Geom. 16, 357-370 (1998).
• [197] Salvai, M.: Density of periodic geodesics in the unit tangent bundle of a compact hyperbolic surface, Rev. Uni. Mat. Argentina 41, 99-105 (1999).
• [198] Salvai, M.: On the geometry at inifinity of the universal covering of Sl(2,R), Rend. Sem. Mat. Univ. Padova 104, 91-108 (2000).
• [199] Sasaki, S.: On the differential geometry of tangent bundles of Riemannian manifolds, Tôhoku Math. J. 10, (3), 338-354 (1958).
• [200] Sasaki, S.: On the differential geometry of tangent bundles of Riemannian manifolds II, Tôhoku Math. J. 14, (2), 146–155 (1962).
• [201] Sasaki, S.: Notes on my mathematical works, Shigeo Sasaki. Selected Papers (S. Tachibana ed), Kinokuniya, 3–30, 1985.
• [202] Sasaki, S., Hatakeyama, Y.: On differential manifolds with certain structures which are closely related to almost contact structure II, Tôhoku Math. J. (2) 13, 282-294 (1961).
• [203] Schlarb, M.: Covariant derivatives on homogeneous spaces. Horizontal lifts and parallel transport, J. Geom. Anal. 34 (5), article number 150, 43 p., (2024).
• [204] Sekigawa, K.: Notes on some curvature homogeneous spaces, Tensor, New Ser. 29, 255-258 (1975).
• [205] Sekigawa, K.: 3-dimensional homogeneous Riemannian manifolds. I, Sci. Rep. Niigata Univ., Ser. A 14, 5-14 (1977).
• [206] Sekigawa, K.: 3-dimensional homogeneous Riemannian manifolds. II, Sci. Rep. Niigata Univ., Ser. A 15, 71-78 (1978).
• [207] Sekigawa, K.: Notes on homogeneous almost Hermitian manifolds, Hokkaido Math. J. 7, 206-213 (1978).
• [208] Strichartz, R. S.:, Sub-Riemannian geometry, J. Differ. Geom. 24, 221-263 (1986). Correction ibid. 30 (2), 595-596 (1989).
• [209] Takahashi, Toshio: Sasakian ϕ-symmetric spaces, Tôhoku Math. J. (2) 29, 91-113 (1977).
• [210] Takahashi, Tsunoro: An isometric immersion of a homogeneous Riemannian manifold of dimension 3 in the hyperbolic space, J. Math. Soc. Japan 23, 649–661 (1971).
• [211] Tamaru, H.: Riemannian g. o. spaces fibered over irreducible symmetric spaces, Osaka J. Math. 36, 835-851 (1999).
• [212] Tanaka, N.: A Differential Geometric Study on Strongly Pseudo-Convex Manifolds, Lecture in Math. Kyoto Univ. 9, Kinokuniya Book Store (1975).
• [213] Tanaka, N.: On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections, Japan. J. Math. 2, 131–190 (1976).
• [214] Tanno, S.: Some transformations on manifolds with almost contact and contact metric structures I, II, Tôhoku Math. J. (2) 15, 140-147, 322-331 (1963).
• [215] Tanno, S.: A theorem on regular vector fields and its applications to almost contact structures, Tôhoku Math. J. (2) 17 (3), 235-238 (1965) .
• [216] Tanno, S.: Locally symmetric K-contact Riemannian manifolds, Proc. Japan Acad. 43, 581-583 (1967).
• [217] Tanno, S.: The automorphism groups of almost contact Riemannian manifolds, Tôhoku Math. J. (2) 21, 21-38 (1969).
• [218] Tanno, S.: Sasakian manifolds with constant φ-holomorphic sectional curvature, Tôhoku Math. J. (2) 21, 501-507 (1969).
• [219] Tanno, S.: Variational problems on contact Riemannian manifolds, Trans, Am. Math. Soc. 314 (1), 349-379 (1989).
• [220] Thurston, W. M.: Three-dimensional Geometry and Topology I (S. Levy ed.), Princeton Math. Series. 35, (1997).
• [221] Tojo, K.: Kähler C-spaces and k-symmetric spaces, Osaka J. Math. 34 (4), 803-820 (1997).
• [222] Tricerri, F., Vanhecke, L.: Homogeneous Structures on Riemannian Manifolds, Lecture Notes Series, London Math. Soc. 52, Cambridge Univ. Press (1983).
• [223] Tricerri, F., Vanhecke, L.: Naturally reductive homogeneous spaces and generalized Heisenberg groups, Compositio Math. 52, 389-408 (1984).
• [224] Vanhecke, L., Willmore, T.J.: Interaction of tubes and spheres, Math. Ann. 263, 31-42 (1983).
• [225] Vezzoni, L.: Connections on contact manifolds and contact twistor space, Israel J. Math. 178, 253-267 (2010).
• [226] Vranceanu, G.: Lecons de Geometrie Differentielle I, Ed. Acad. Rep. Roum, Bucarest, (1947).
• [227] Watanabe, Y., Tricerri, F.: Characterizations of ϕ-symmetric spaces in terms of the canonical connection, C. R. Math. Rep. Acad. Sci. Canada 15 (2-3), 61-66 (1993).
• [228] Webster, S. M.: Pseudohermitian structures on a real hypersurface, J. Differential Geom. 13, 25-41 (1978).
• [229] Witte, D.: Cocompact subgroups of semisimple Lie groups, Lie algebra and related topics (Madison, WI, 1988), 309–313, Contemp. Math. 110, Am. Math. Soc., Providence, RI, 1990.