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Left-Invariant Minimal Unit Vector Fields on the Solvable Lie Group

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Abstract

Božek (1980) has introduced a class of solvable Lie groups Gn with arbitrary odd dimension to construct irreducible generalized symmetric Riemannian space such that the identity component of its full isometry group is solvable. In this article, the authors provide the set of all left-invariant minimal unit vector fields on the solvable Lie group Gn, and give the relationships between the minimal unit vector fields and the geodesic vector fields, the strongly normal unit vectors respectively.

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References

  1. Aghasi, M. and Nasehi, M., Some geometrical properties of a five-dimensional solvable Lie group, Differ. Geom. Dyn. Syst., 15, 2013, 1–12.

    MathSciNet  MATH  Google Scholar 

  2. Aghasi, M. and Nasehi, M., On the geometrical properties of solvable Lie groups, Adv. Geom., 15(4), 2015, 507–517.

    Article  MathSciNet  MATH  Google Scholar 

  3. Aghasi, M. and Nasehi, M., On homogeneous Randers spaces with Douglas or naturally reductive metrics, Differ. Geom. Dyn. Syst., 17, 2015, 1–12.

    MathSciNet  MATH  Google Scholar 

  4. Boeckx, E. and Vanhecke, L., Harmonic and minimal radial vector fields, Acta Math. Hungar., 90(4), 2001, 317–331.

    Article  MathSciNet  MATH  Google Scholar 

  5. Božek, M., Existence of generalized symmetric Riemannian spaces with solvable isometry grooup, Časopis Pěst. Mat., 105, 1980, 368–384.

    Article  MathSciNet  MATH  Google Scholar 

  6. Calvaruso, G., Kowalski O. and Marinosci, R., Homogeneous geodesics in solvable Lie groups, Acta Math. Hungar., 101, 2003, 313–322.

    Article  MathSciNet  MATH  Google Scholar 

  7. Gil-Medriano, O. and Llinares-Fuster, E., Minimal unit vector fields, Tohoku Math. J., 54(2), 2002, 77–93.

    MathSciNet  MATH  Google Scholar 

  8. Gluck, H. and Ziller, W., On the volume of a unit vector field on the three-sphere, Comment. Math. Helv., 61(2), 1986, 177–192.

    Article  MathSciNet  MATH  Google Scholar 

  9. Gonzales-Davila, J. C. and Vanhecke, L., Examples of minimal unit vector fields, Ann. Global Anal. Geom., 18(3–4), 2000, 385–404.

    Article  MathSciNet  MATH  Google Scholar 

  10. Johnson, D. L., Volumes of flows, Proc. Amer. Math. Soc., 104, 1988, 923–931.

    Article  MathSciNet  MATH  Google Scholar 

  11. Johnson, D. L. and Smith, P., Regularity of volume-minimizing graphs, Indiana Univ. Math. J., 44, 1995, 45–85.

    Article  MathSciNet  MATH  Google Scholar 

  12. Milnor, J., Curvatures of left invariant metrics on Lie groups, Adv. Math., 21(3), 1976, 293–329.

    Article  MathSciNet  MATH  Google Scholar 

  13. Pedersen, S. L., Volumes of vector fields on spheres, Trans. Amer. Math. Soc., 336, 1993, 69–78.

    Article  MathSciNet  MATH  Google Scholar 

  14. Reznikov, A. G., Lower bounds on volumes of vector fields, Arch. Math., 58, 1992, 509–513.

    Article  MathSciNet  MATH  Google Scholar 

  15. Salvai, M., On the volume of unit vector fields on a compact semisimple Lie group, J. Lie Theory., 13(2), 2003, 457–464.

    MathSciNet  MATH  Google Scholar 

  16. Tsukada, K. and Vanhecke, L., Invariant minimal unit vector fields on Lie groups, Period. Math. Hungar., 40(2), 2000, 123–133.

    Article  MathSciNet  MATH  Google Scholar 

  17. Yi, S., Left-invariant minimal unit vector fields on a Lie group of constant negative sectional curvature, Bull. Korean Math. Soc., 46(4), 2009, 713–720.

    Article  MathSciNet  MATH  Google Scholar 

  18. Yi, S., Left-invariant minimal unit vector fields on the semi-direct product ℝn × p ℝ, Bull. Korean Math. Soc., 47(5), 2010, 951–960.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, Shandong, 266590, China

    Shaoxiang Zhang

  2. School of Microelectronics and Data Science, Anhui University of Technology, Maanshan, Anhui, 243032, China

    Ju Tan

Authors
  1. Shaoxiang Zhang
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  2. Ju Tan
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Corresponding author

Correspondence to Ju Tan.

Additional information

This work was supported by the National Natural Science Foundation of China (Nos. 12001007, 12201358), the Natural Science Foundation of Shandong Province (No. ZR2021QA051), the Natural Science Foundation of Anhui Province (No. 1908085QA03) and Starting Research Funds of Shandong University of Science and Technology (Nos. 0104060511817, 0104060540626).

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Zhang, S., Tan, J. Left-Invariant Minimal Unit Vector Fields on the Solvable Lie Group. Chin. Ann. Math. Ser. B 44, 67–80 (2023). https://doi.org/10.1007/s11401-023-0005-1

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  • DOI: https://doi.org/10.1007/s11401-023-0005-1

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