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Year 2024, Volume: 17 Issue: 1, 106 - 136, 23.04.2024

Jun-ichi Inoguchı

https://doi.org/10.36890/iejg.1429782
Cited By: 1

Abstract

References

  • [1] Alekseevskii, D.V.: Homogeneous Riemannian spaces of negative curvature. Mat. Sb. 96(138), 87–109 (1975).
  • [2] Andrada, A., Barberis, M. L., Dotti, I. G., Ovando, G. P.: Product structures on four dimensional solvable Lie algebras. Homol. Homotopy Appl 7, 9-37 (2005)
  • [3] Arvanitoyeorgos, A., Panagiotis Souris, N.: Two-step homogeneous geodesics in homogeneous spaces. Taiwanese J. Math. 20 (6), 1313- 1333 (2016)
  • [4] Ateş, O., Munteanu, M. I., Periodic J-trajectories on S3 × R. J. Geom. Phys. 133, 141-152 (2018)
  • [5] Berestovskii, V., Nikonorov, Y.: Riemannian Manifolds and Homogeneous Geodesics, Springer Cham., (2020).
  • [6] Biggs, R., Remsing, C. C.: On the classification of real four-dimensional Lie groups. J. Lie Theory 26 (4), 1001-1035 (2016).
  • [7] Chen, B.-Y.: Geometry of Submanifolds and its Applications. Science University of Tokyo, 96 p. (1981)
  • [8] Chen, B.-Y.: Circles in compact homogeneous Riemannian spaces and immersions of finite type. Glasgow Math. J. 44, 93-102 (2002).
  • [9] Chen, B.-Y., Leung, P., Nagano, T.: Totally geodesic submanifolds of symmetric spaces, III. arXiv:1307.7325 [math.DG] (2013).
  • [10] Chen, B.-Y., Maeda, S.: Extrinsic characterizations of circles in a complex projective space imbedded in a Euclidean space. Tokyo J. Math. 19 (1): 169-185 (1996).
  • [11] Chen, B.-Y., Nagano, T.: Totally geodesic submanifolds of symmetric spaces. I. Duke Math. J. 44, 745-755 (1977).
  • [12] Chen, B.-Y., Nagano, T.: Totally geodesic submanifolds of symmetric spaces. II. Duke Math. J. 45, 405-425 (1978).
  • [13] Chen, B.-Y., Piccinni, P.: The canonical foliations of a locally conformal Kähler manifold. Ann. Mat. Pura Appl. (4) 141, 289-305 (1985).
  • [14] D’Atri, J. E. Ziller, W.: Naturally Reductive Metrics and Einstein Metrics on Compact Lie Groups. Mem. Amer. Math. Soc. 215 (1979).
  • [15] D’haene, M.: Submanifolds of the Thurston geometry Sol40 . Master Thesis, KU Leuven (2020).
  • [16] D’haene, M.: Thurston geometries in dimension four from a Riemannian perspective. arXiv:2401.05977v1 [math.DG] (2024).
  • [17] D’haene, M., Inoguchi, J., Van der Veken, J.: Parallel and totally umbilical hypersurfaces of the four-dimensional Thurston geometry Sol40 , Math. Nach., to appear.
  • [18] Djellali, N., Hasni, A., Cherif, A. M., Belkhelfa, M.: Classification of Codazzi and note on minimal hypersurfaces in Nil4. Internat. Elect. J. Geom. 16(2), 707-714 (2023).
  • [19] Dohira, R.: Geodesics in reductive homogeneous spaces. Tsukuba J. Math. 19 (1), 233-243 (1995).
  • [20] Erjavec, Z., Inoguchi, J.: Magnetic curves in H3 × R. J. Korean Math. Soc. 58 (6), 1501-1511 (2021).
  • [21] Erjavec, Z., Inoguchi, J.: J-trajectories in 4-dimensional solvable Lie groups Sol40 . Math. Phys. Anal. Geom. 25, Article number 8 (2022).
  • [22] Erjavec, Z., Inoguchi, J.: Minimal submanifolds in Sol40 . J. Geom. Anal. 33 (9), Paper No. 274, 39 p. (2023).
  • [23] Erjavec, Z., Inoguchi, J.: J–trajectories in 4-dimensional solvable Lie group Sol41 . J. Nonlinear Sci. 33 (6), Paper No. 111, 37 p. (2023).
  • [24] Erjavec, Z., Inoguchi, J.: Minimal submanifolds in Sol41 . Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 117 (4), Paper No. 156, 36 p. (2023).
  • [25] Erjavec, Z., Inoguchi, J.: Minimal submanifolds in H3 × R. submitted.
  • [26] Erjavec, Z., Inoguchi, J.: Geodesics and magnetic curves in the 4-dim almost Kähler model space F4. Complex Manifolds, to appear.
  • [27] Filipkiewicz, R.: Four dimensional geometries. Ph. D. Thesis, University of Warwick (1983)
  • [28] Gordon, C. S.: Naturally reductive homogeneous Riemannian manifolds. Canadian J. Math. 37 (3), 467–487 (1985)
  • [29] Gordon, C. S.: Homogeneous Riemannian manifolds whose geodesics are orbits. Topics in Geometry. In memory of Joseph D’Atri (Gindikin, Simon ed.), Boston, MA: Birkhäuser. Prog. Nonlinear Differ. Equ. Appl. 20, 155-174 (1996)
  • [30] Gordon, C., Wilson, E.N.: Isometry groups of Riemannian solvmanifolds. Trans. Am. Math. Soc. 307, 245–269 (1988)
  • [31] 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)
  • [32] 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)
  • [33] Hashinaga, T.: On the minimality of the corresponding submanifolds to four-dimensional solvsolitons, Hiroshima Math. J. 44 (2), 173-191 (2014)
  • [34] Hashinaga, T., Tamaru, H., Terada, K.: Milnor-type theorems for left-invariant Riemannian metrics on Lie groups. J. Math. Soc. Japan 68(2), 669-684 (2016)
  • [35] Hillman, J. A.: Geometries and infrasolvmanifolds in dimension 4, Geom. Dedicata 129, 57-72 (2007)
  • [36] Hillman, J. A.: Four-manifolds, Geometries and Knots. Geom. Topol. Monogr. Vol. 5 (2002). Revised version. arXiv:math/0212142v3 [math.GT]
  • [37] Inoguchi, J.: J-trajectories in locally conformal Kähler manifolds with parallel anti-Lee field. Int. Electron. J. Geom. 13 (2), 30-44 (2020)
  • [38] Inoue, M.: On surfaces of class VII0. Invent. Math. 24, 269-310 (1974)
  • [39] Jo, J. H., Lee, J. B.: Nielsen type numbers and homotopy minimal periods for maps on solvmanifolds with Sol41 -geometry. Fixed Point Theory and Applications 175, 1-15 (2015)
  • [40] Kajzer, V. V.: Conjugate points of left-invariant metrics on Lie groups. Soviet Math. 34, 32-44 (1990)
  • [41] Klein, S.: Totally geodesic submanifolds of the complex quadric. Differential Geom. Appl. 26, 79-96 (2008)
  • [42] Klein, S.: Totally geodesic submanifolds of the complex and the quaternionic 2-Grassmannians. Trans. Amer. Math. Soc. 361, 4927-4967 (2009)
  • [43] Klein, S.: Reconstructing the geometric structure of a Riemannian symmetric space from its Satake diagram, Geom. Dedicata 138 (2009), 25-50 (2009)
  • [44] Klein, S.: Totally geodesic subanifolds of the exceptional Riemannian symmetric spaces of rank 2. Osaka J. Math. 47 (2010), 1077-1157 (2010)
  • [45] Kodama, H., Takahara, A., Tamaru, H.: The space of left-invariant metrics on a Lie group up to isometry and scaling. Manuscripta Math. 135(1-2), 229-243 (2011)
  • [46] Kostant, B.: Holonomy and the Lie algebra of infinitesimal motions of a Riemannian manifold. Trans. Amer. Math. Soc. 80, 528-542 (1955)
  • [47] Kowalski, O.: Generalized Symmetric Spaces. Lecture Notes in Mathematics. 805. Springer-Verlag, (1980)
  • [48] Kowalski, O., Nikˇcevi´c, S., Vlášek, Z.: Homogeneous geodesics in homogeneous Riemannian manifolds (examples), in: Geometry and Topology of Submanifolds, (Beijing/Berlin, 1999), World Sci. Publishing co., River Edge, NJ, 2000, pp. 104–112.
  • [49] Kowalski, O., Szenthe, J.: On the existence of homogeneous geodesics in homogeneous Riemannian manifolds. Geom. Dedicata 81 (1-3), 209-214 (2000). 84 (1-3), 331-332 (2001)
  • [50] Kowalski, O., Tricerri, F.: Riemannian manifolds of dimension N ≤ 4 admitting a homogeneous structure of class T2. Conf. Semin. Mat. Univ. Bari 222, 24 p. (1987).
  • [51] Kowalski, O., Vanhecke, L.: Four-dimensional naturally reductive homogeneous spaces. Differential geometry on homogeneous spaces, Conf. Torino/Italy 1983, Rend. Semin. Mat., Torino, Fasc. Spec., 223-232 (1983).
  • [52] Kowalski, O., Vanhecke, L.: Riemannian manifolds with homogeneous geodesics. Boll. Unione Mat. Ital., VII. Ser., B 5 (1), 189-246 (1991).
  • [53] Lauret, J.: Homogeneous nilmanifolds of dimensions 3 and 4. Geom. Dedicata 68, 145–155 (1997)
  • [54] Lauret, J.: Degenerations of Lie algebras and geometry of Lie groups. Differential Geom. Appl. 18(2), 177-194 (2003)
  • [55] Lee, J. B., Lee, K. B., Shin, J., Yi, S.: Unimodular groups of type R3 ⋊ R. J. Korean Math. Soc, 44 (5), 1121-1137 (2007)
  • [56] MacCallum, M. A. H.: On the classification of the real four-dimensional Lie algebras. On Einstein’s Path. (A. Harvey ed.), Springer, New York, 299-317, (1999)
  • [57] Magnin, L.: Sur les algébres de Lie nilpotents de dimension ≤ 7. J. Geom. Phys. 3(1), 119-144 (1986)
  • [58] Manzano, J. M., Torralbo, F., Van der Veken, J.: Parallel mean curvature surfaces in four-dimensional homogeneous spaces. Proceedings Book of International Workshop on Theory of Submanifolds 1, 57-78 (2016)
  • [59] Marinosci, R. A.: Homogeneous geodesics in a three-dimensional Lie group. Comment. Math. Univ. Carolinae 43(2), 261-270 (2002) [60] Mashimo, K., Tojo, K.: Circles in Riemannian symmetric spaces. Kodai Math. J. 22(1), 1-14 (1999)
  • [61] Matsushita, Y.: Geometric structures in four-dimension and almost Hermitian structures. AIP Conference Proceedings 1340(1), 66-80 (2011)
  • [62] Nikolayevsky, Y.: Totally geodesic hypersurfaces of homogeneous spaces. Isr. J. Math. 207, 361-375 (2015)
  • [63] Oguro, T., Sekigawa, K.: Almost Kähler structures on the Riemannian product of a 3-dimensional hyperbolic space and a real line. Tsukuba J. Math. 20(1), 151–161 (1996)
  • [64] Otsuki, T., Tashiro, Y.: On curves in Kaehlerian spaces. Math. J. Okayama Univ. 4, 57–78 (1954)
  • [65] Ovando, G.: Invariant complex structures on solvable real Lie groups, Manuscripta Math. 103, 19-30 (2000)
  • [66] Ovando, G.: Complex, symplectic and Kähler structures on four dimensional Lie groups, Rev. Un. Mat. Argentina 45(2), 55-67 (2004)
  • [67] Patera, J., Winternitz, P.: Subalgebras of real three- and four-dimensional Lie algebras. J. Math. Phys. 18, 1449-1455 (1977)
  • [68] Sawai, H.: Locally conformal Kähler structures on compact solvmanifolds. Osaka J. Math. 49, 1087-1102 (2012)
  • [69] Shin, J.: Isometry groups of unimodular simply connected 3-dimensional Lie groups. Geom. Dedicata 65, 267–290 (1997).
  • [70] Snow, J. E.: Invariant complex structures on four-dimensional solvable real Lie groups. Manuscripta Math. 66, 397-412 (1990)
  • [71] Şukiloviç, T.: Classification of left invariant metrics on 4-dimensional solvable Lie groups. Theoret. Appl. Mech. 47(2), 181-204, 2020
  • [72] Thurston, W. M.: Three-dimensional Geometry and Topology I, Princeton Math. Series, vol. 35 (S. Levy ed.), 1997.
  • [73] Tojo, K.: Totally geodesic submanifolds of naturally reductive homogeneous spaces. Tsukuba J. Math. 20(1), 181-190 (1996)
  • [74] Tojo, K.: Normal homogeneous spaces admitting totally geodesic hypersurfaces. J. Math. Soc. Japan 49(4), 781-815 (1997)
  • [75] Tricerri, F.: Some examples of locally conformal Kähler manifolds. Rend. Sem. Mat. Univ. Politec. Torino 40, 81-92 (1982)
  • [76] Tricerri, F., Vanhecke, L.: Homogeneous Structures on Riemannian Manifolds, London Mathematical Society Lecture Note Series 83, Cambridge University Press, 1983.
  • [77] Tsukada, K.: Totally geodesic hypersurfaces of naturally reductive homogeneous spaces. Osaka J. Math. 33(3), 697-707 (1996).
  • [78] Van Thuong, S.: Classification, cobordism, and curvature of four-dimensional infra-solvmanifolds. Ph. D. Thesis, University of Oklahoma, (2014)
  • [79] Van Thuong, S.: All 4-dimensional infra-solvmanifolds are boundaries. Geom Dedicata 176, 315–328 (2015)
  • [80] Van Thuong, S.: Metrics on 4-dimensional unimodular Lie groups. Ann. Glob. Anal. Geom. 51, 109-128 (2017)
  • [81] Van Thuong, S.: Classification of closed manifolds with Sol41 -geometry. Geom. Dedicata 199, 373-397 (2019)
  • [82] Vinberg, È. B.: Invariant linear connections in a homogeneous space (Russian). Trudy Moskov. Mat. Obšˇc. 9, 191–210 (1960)
  • [83] Wall, C.T.C.: Geometries and geometric structures in real dimension 4 and complex dimension 2. Geometry and Topology, Lecture Notes in Math. 1167 268-292 (1985)
  • [84] Wall, C.T.C.: Geometric structures on compact complex analytic surfaces, Topology 25(2), 119-153 (1986)
  • [85] Wang, H.-C.: Discrete nilpotent subgroups of Lie groups. J. Differ. Geom. 3, 481-492 (1969).
  • [86] Wilson, E. N.: Isometry groups on homogeneous nilmanifolds. Geom Dedicata 12, 337–346 (1982)
  • [87] Yoo, W. S.: On the crystallographic group of Solm,n. J. Chungcheong Math. Soc. 34(1), 17-26 (2021)

Homogeneous Geodesics of $4$-dimensional Solvable Lie Groups

Year 2024, Volume: 17 Issue: 1, 106 - 136, 23.04.2024

Jun-ichi Inoguchı

https://doi.org/10.36890/iejg.1429782
Cited By: 1

Abstract

We study homogeneous geodesics in $4$-dimensional solvable Lie groups $\mathrm{Sol}_0^4$,
$\mathrm{Sol}_1^4$, $\mathrm{Sol}_{m,n}$ and $\mathrm{Nil}_4$.

Keywords

Homogeneous geodesics solvable Lie group 4-dimensional geometry

References

  • [1] Alekseevskii, D.V.: Homogeneous Riemannian spaces of negative curvature. Mat. Sb. 96(138), 87–109 (1975).
  • [2] Andrada, A., Barberis, M. L., Dotti, I. G., Ovando, G. P.: Product structures on four dimensional solvable Lie algebras. Homol. Homotopy Appl 7, 9-37 (2005)
  • [3] Arvanitoyeorgos, A., Panagiotis Souris, N.: Two-step homogeneous geodesics in homogeneous spaces. Taiwanese J. Math. 20 (6), 1313- 1333 (2016)
  • [4] Ateş, O., Munteanu, M. I., Periodic J-trajectories on S3 × R. J. Geom. Phys. 133, 141-152 (2018)
  • [5] Berestovskii, V., Nikonorov, Y.: Riemannian Manifolds and Homogeneous Geodesics, Springer Cham., (2020).
  • [6] Biggs, R., Remsing, C. C.: On the classification of real four-dimensional Lie groups. J. Lie Theory 26 (4), 1001-1035 (2016).
  • [7] Chen, B.-Y.: Geometry of Submanifolds and its Applications. Science University of Tokyo, 96 p. (1981)
  • [8] Chen, B.-Y.: Circles in compact homogeneous Riemannian spaces and immersions of finite type. Glasgow Math. J. 44, 93-102 (2002).
  • [9] Chen, B.-Y., Leung, P., Nagano, T.: Totally geodesic submanifolds of symmetric spaces, III. arXiv:1307.7325 [math.DG] (2013).
  • [10] Chen, B.-Y., Maeda, S.: Extrinsic characterizations of circles in a complex projective space imbedded in a Euclidean space. Tokyo J. Math. 19 (1): 169-185 (1996).
  • [11] Chen, B.-Y., Nagano, T.: Totally geodesic submanifolds of symmetric spaces. I. Duke Math. J. 44, 745-755 (1977).
  • [12] Chen, B.-Y., Nagano, T.: Totally geodesic submanifolds of symmetric spaces. II. Duke Math. J. 45, 405-425 (1978).
  • [13] Chen, B.-Y., Piccinni, P.: The canonical foliations of a locally conformal Kähler manifold. Ann. Mat. Pura Appl. (4) 141, 289-305 (1985).
  • [14] D’Atri, J. E. Ziller, W.: Naturally Reductive Metrics and Einstein Metrics on Compact Lie Groups. Mem. Amer. Math. Soc. 215 (1979).
  • [15] D’haene, M.: Submanifolds of the Thurston geometry Sol40 . Master Thesis, KU Leuven (2020).
  • [16] D’haene, M.: Thurston geometries in dimension four from a Riemannian perspective. arXiv:2401.05977v1 [math.DG] (2024).
  • [17] D’haene, M., Inoguchi, J., Van der Veken, J.: Parallel and totally umbilical hypersurfaces of the four-dimensional Thurston geometry Sol40 , Math. Nach., to appear.
  • [18] Djellali, N., Hasni, A., Cherif, A. M., Belkhelfa, M.: Classification of Codazzi and note on minimal hypersurfaces in Nil4. Internat. Elect. J. Geom. 16(2), 707-714 (2023).
  • [19] Dohira, R.: Geodesics in reductive homogeneous spaces. Tsukuba J. Math. 19 (1), 233-243 (1995).
  • [20] Erjavec, Z., Inoguchi, J.: Magnetic curves in H3 × R. J. Korean Math. Soc. 58 (6), 1501-1511 (2021).
  • [21] Erjavec, Z., Inoguchi, J.: J-trajectories in 4-dimensional solvable Lie groups Sol40 . Math. Phys. Anal. Geom. 25, Article number 8 (2022).
  • [22] Erjavec, Z., Inoguchi, J.: Minimal submanifolds in Sol40 . J. Geom. Anal. 33 (9), Paper No. 274, 39 p. (2023).
  • [23] Erjavec, Z., Inoguchi, J.: J–trajectories in 4-dimensional solvable Lie group Sol41 . J. Nonlinear Sci. 33 (6), Paper No. 111, 37 p. (2023).
  • [24] Erjavec, Z., Inoguchi, J.: Minimal submanifolds in Sol41 . Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 117 (4), Paper No. 156, 36 p. (2023).
  • [25] Erjavec, Z., Inoguchi, J.: Minimal submanifolds in H3 × R. submitted.
  • [26] Erjavec, Z., Inoguchi, J.: Geodesics and magnetic curves in the 4-dim almost Kähler model space F4. Complex Manifolds, to appear.
  • [27] Filipkiewicz, R.: Four dimensional geometries. Ph. D. Thesis, University of Warwick (1983)
  • [28] Gordon, C. S.: Naturally reductive homogeneous Riemannian manifolds. Canadian J. Math. 37 (3), 467–487 (1985)
  • [29] Gordon, C. S.: Homogeneous Riemannian manifolds whose geodesics are orbits. Topics in Geometry. In memory of Joseph D’Atri (Gindikin, Simon ed.), Boston, MA: Birkhäuser. Prog. Nonlinear Differ. Equ. Appl. 20, 155-174 (1996)
  • [30] Gordon, C., Wilson, E.N.: Isometry groups of Riemannian solvmanifolds. Trans. Am. Math. Soc. 307, 245–269 (1988)
  • [31] 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)
  • [32] 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)
  • [33] Hashinaga, T.: On the minimality of the corresponding submanifolds to four-dimensional solvsolitons, Hiroshima Math. J. 44 (2), 173-191 (2014)
  • [34] Hashinaga, T., Tamaru, H., Terada, K.: Milnor-type theorems for left-invariant Riemannian metrics on Lie groups. J. Math. Soc. Japan 68(2), 669-684 (2016)
  • [35] Hillman, J. A.: Geometries and infrasolvmanifolds in dimension 4, Geom. Dedicata 129, 57-72 (2007)
  • [36] Hillman, J. A.: Four-manifolds, Geometries and Knots. Geom. Topol. Monogr. Vol. 5 (2002). Revised version. arXiv:math/0212142v3 [math.GT]
  • [37] Inoguchi, J.: J-trajectories in locally conformal Kähler manifolds with parallel anti-Lee field. Int. Electron. J. Geom. 13 (2), 30-44 (2020)
  • [38] Inoue, M.: On surfaces of class VII0. Invent. Math. 24, 269-310 (1974)
  • [39] Jo, J. H., Lee, J. B.: Nielsen type numbers and homotopy minimal periods for maps on solvmanifolds with Sol41 -geometry. Fixed Point Theory and Applications 175, 1-15 (2015)
  • [40] Kajzer, V. V.: Conjugate points of left-invariant metrics on Lie groups. Soviet Math. 34, 32-44 (1990)
  • [41] Klein, S.: Totally geodesic submanifolds of the complex quadric. Differential Geom. Appl. 26, 79-96 (2008)
  • [42] Klein, S.: Totally geodesic submanifolds of the complex and the quaternionic 2-Grassmannians. Trans. Amer. Math. Soc. 361, 4927-4967 (2009)
  • [43] Klein, S.: Reconstructing the geometric structure of a Riemannian symmetric space from its Satake diagram, Geom. Dedicata 138 (2009), 25-50 (2009)
  • [44] Klein, S.: Totally geodesic subanifolds of the exceptional Riemannian symmetric spaces of rank 2. Osaka J. Math. 47 (2010), 1077-1157 (2010)
  • [45] Kodama, H., Takahara, A., Tamaru, H.: The space of left-invariant metrics on a Lie group up to isometry and scaling. Manuscripta Math. 135(1-2), 229-243 (2011)
  • [46] Kostant, B.: Holonomy and the Lie algebra of infinitesimal motions of a Riemannian manifold. Trans. Amer. Math. Soc. 80, 528-542 (1955)
  • [47] Kowalski, O.: Generalized Symmetric Spaces. Lecture Notes in Mathematics. 805. Springer-Verlag, (1980)
  • [48] Kowalski, O., Nikˇcevi´c, S., Vlášek, Z.: Homogeneous geodesics in homogeneous Riemannian manifolds (examples), in: Geometry and Topology of Submanifolds, (Beijing/Berlin, 1999), World Sci. Publishing co., River Edge, NJ, 2000, pp. 104–112.
  • [49] Kowalski, O., Szenthe, J.: On the existence of homogeneous geodesics in homogeneous Riemannian manifolds. Geom. Dedicata 81 (1-3), 209-214 (2000). 84 (1-3), 331-332 (2001)
  • [50] Kowalski, O., Tricerri, F.: Riemannian manifolds of dimension N ≤ 4 admitting a homogeneous structure of class T2. Conf. Semin. Mat. Univ. Bari 222, 24 p. (1987).
  • [51] Kowalski, O., Vanhecke, L.: Four-dimensional naturally reductive homogeneous spaces. Differential geometry on homogeneous spaces, Conf. Torino/Italy 1983, Rend. Semin. Mat., Torino, Fasc. Spec., 223-232 (1983).
  • [52] Kowalski, O., Vanhecke, L.: Riemannian manifolds with homogeneous geodesics. Boll. Unione Mat. Ital., VII. Ser., B 5 (1), 189-246 (1991).
  • [53] Lauret, J.: Homogeneous nilmanifolds of dimensions 3 and 4. Geom. Dedicata 68, 145–155 (1997)
  • [54] Lauret, J.: Degenerations of Lie algebras and geometry of Lie groups. Differential Geom. Appl. 18(2), 177-194 (2003)
  • [55] Lee, J. B., Lee, K. B., Shin, J., Yi, S.: Unimodular groups of type R3 ⋊ R. J. Korean Math. Soc, 44 (5), 1121-1137 (2007)
  • [56] MacCallum, M. A. H.: On the classification of the real four-dimensional Lie algebras. On Einstein’s Path. (A. Harvey ed.), Springer, New York, 299-317, (1999)
  • [57] Magnin, L.: Sur les algébres de Lie nilpotents de dimension ≤ 7. J. Geom. Phys. 3(1), 119-144 (1986)
  • [58] Manzano, J. M., Torralbo, F., Van der Veken, J.: Parallel mean curvature surfaces in four-dimensional homogeneous spaces. Proceedings Book of International Workshop on Theory of Submanifolds 1, 57-78 (2016)
  • [59] Marinosci, R. A.: Homogeneous geodesics in a three-dimensional Lie group. Comment. Math. Univ. Carolinae 43(2), 261-270 (2002) [60] Mashimo, K., Tojo, K.: Circles in Riemannian symmetric spaces. Kodai Math. J. 22(1), 1-14 (1999)
  • [61] Matsushita, Y.: Geometric structures in four-dimension and almost Hermitian structures. AIP Conference Proceedings 1340(1), 66-80 (2011)
  • [62] Nikolayevsky, Y.: Totally geodesic hypersurfaces of homogeneous spaces. Isr. J. Math. 207, 361-375 (2015)
  • [63] Oguro, T., Sekigawa, K.: Almost Kähler structures on the Riemannian product of a 3-dimensional hyperbolic space and a real line. Tsukuba J. Math. 20(1), 151–161 (1996)
  • [64] Otsuki, T., Tashiro, Y.: On curves in Kaehlerian spaces. Math. J. Okayama Univ. 4, 57–78 (1954)
  • [65] Ovando, G.: Invariant complex structures on solvable real Lie groups, Manuscripta Math. 103, 19-30 (2000)
  • [66] Ovando, G.: Complex, symplectic and Kähler structures on four dimensional Lie groups, Rev. Un. Mat. Argentina 45(2), 55-67 (2004)
  • [67] Patera, J., Winternitz, P.: Subalgebras of real three- and four-dimensional Lie algebras. J. Math. Phys. 18, 1449-1455 (1977)
  • [68] Sawai, H.: Locally conformal Kähler structures on compact solvmanifolds. Osaka J. Math. 49, 1087-1102 (2012)
  • [69] Shin, J.: Isometry groups of unimodular simply connected 3-dimensional Lie groups. Geom. Dedicata 65, 267–290 (1997).
  • [70] Snow, J. E.: Invariant complex structures on four-dimensional solvable real Lie groups. Manuscripta Math. 66, 397-412 (1990)
  • [71] Şukiloviç, T.: Classification of left invariant metrics on 4-dimensional solvable Lie groups. Theoret. Appl. Mech. 47(2), 181-204, 2020
  • [72] Thurston, W. M.: Three-dimensional Geometry and Topology I, Princeton Math. Series, vol. 35 (S. Levy ed.), 1997.
  • [73] Tojo, K.: Totally geodesic submanifolds of naturally reductive homogeneous spaces. Tsukuba J. Math. 20(1), 181-190 (1996)
  • [74] Tojo, K.: Normal homogeneous spaces admitting totally geodesic hypersurfaces. J. Math. Soc. Japan 49(4), 781-815 (1997)
  • [75] Tricerri, F.: Some examples of locally conformal Kähler manifolds. Rend. Sem. Mat. Univ. Politec. Torino 40, 81-92 (1982)
  • [76] Tricerri, F., Vanhecke, L.: Homogeneous Structures on Riemannian Manifolds, London Mathematical Society Lecture Note Series 83, Cambridge University Press, 1983.
  • [77] Tsukada, K.: Totally geodesic hypersurfaces of naturally reductive homogeneous spaces. Osaka J. Math. 33(3), 697-707 (1996).
  • [78] Van Thuong, S.: Classification, cobordism, and curvature of four-dimensional infra-solvmanifolds. Ph. D. Thesis, University of Oklahoma, (2014)
  • [79] Van Thuong, S.: All 4-dimensional infra-solvmanifolds are boundaries. Geom Dedicata 176, 315–328 (2015)
  • [80] Van Thuong, S.: Metrics on 4-dimensional unimodular Lie groups. Ann. Glob. Anal. Geom. 51, 109-128 (2017)
  • [81] Van Thuong, S.: Classification of closed manifolds with Sol41 -geometry. Geom. Dedicata 199, 373-397 (2019)
  • [82] Vinberg, È. B.: Invariant linear connections in a homogeneous space (Russian). Trudy Moskov. Mat. Obšˇc. 9, 191–210 (1960)
  • [83] Wall, C.T.C.: Geometries and geometric structures in real dimension 4 and complex dimension 2. Geometry and Topology, Lecture Notes in Math. 1167 268-292 (1985)
  • [84] Wall, C.T.C.: Geometric structures on compact complex analytic surfaces, Topology 25(2), 119-153 (1986)
  • [85] Wang, H.-C.: Discrete nilpotent subgroups of Lie groups. J. Differ. Geom. 3, 481-492 (1969).
  • [86] Wilson, E. N.: Isometry groups on homogeneous nilmanifolds. Geom Dedicata 12, 337–346 (1982)
  • [87] Yoo, W. S.: On the crystallographic group of Solm,n. J. Chungcheong Math. Soc. 34(1), 17-26 (2021)
There are 86 citations in total.

Details

Primary Language English
Subjects Algebraic and Differential Geometry
Journal Section Research Article
Authors

Jun-ichi Inoguchı 0000-0002-6584-5739

Early Pub Date April 5, 2024
Publication Date April 23, 2024
Submission Date February 1, 2024
Acceptance Date March 24, 2024
Published in Issue Year 2024 Volume: 17 Issue: 1

Cite

APA Inoguchı, J.- ichi. (2024). Homogeneous Geodesics of $4$-dimensional Solvable Lie Groups. International Electronic Journal of Geometry, 17(1), 106-136. https://doi.org/10.36890/iejg.1429782
AMA Inoguchı J ichi. Homogeneous Geodesics of $4$-dimensional Solvable Lie Groups. Int. Electron. J. Geom. April 2024;17(1):106-136. doi:10.36890/iejg.1429782
Chicago Inoguchı, Jun-ichi. “Homogeneous Geodesics of $4$-Dimensional Solvable Lie Groups”. International Electronic Journal of Geometry 17, no. 1 (April 2024): 106-36. https://doi.org/10.36890/iejg.1429782.
EndNote Inoguchı J- ichi (April 1, 2024) Homogeneous Geodesics of $4$-dimensional Solvable Lie Groups. International Electronic Journal of Geometry 17 1 106–136.
IEEE J.- ichi Inoguchı, “Homogeneous Geodesics of $4$-dimensional Solvable Lie Groups”, Int. Electron. J. Geom., vol. 17, no. 1, pp. 106–136, 2024, doi: 10.36890/iejg.1429782.
ISNAD Inoguchı, Jun-ichi. “Homogeneous Geodesics of $4$-Dimensional Solvable Lie Groups”. International Electronic Journal of Geometry 17/1 (April2024), 106-136. https://doi.org/10.36890/iejg.1429782.
JAMA Inoguchı J- ichi. Homogeneous Geodesics of $4$-dimensional Solvable Lie Groups. Int. Electron. J. Geom. 2024;17:106–136.
MLA Inoguchı, Jun-ichi. “Homogeneous Geodesics of $4$-Dimensional Solvable Lie Groups”. International Electronic Journal of Geometry, vol. 17, no. 1, 2024, pp. 106-3, doi:10.36890/iejg.1429782.
Vancouver Inoguchı J- ichi. Homogeneous Geodesics of $4$-dimensional Solvable Lie Groups. Int. Electron. J. Geom. 2024;17(1):106-3.

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