DOI QR코드

DOI QR Code

LEFT-INVARIANT MINIMAL UNIT VECTOR FIELDS ON THE SEMI-DIRECT PRODUCT Rn

Yi, Seung-Hun

  • Received : 2009.03.10
  • Accepted : 2009.04.27
  • Published : 2010.09.30

Abstract

We provide the set of left-invariant minimal unit vector fields on the semi-direct product $\mathbb{R}^n\;{\rtimes}_p\mathbb{R}$, where P is a nonsingular diagonal matrix and on the 7 classes of 4-dimensional solvable Lie groups of the form $\mathbb{R}^3\;{\rtimes}_p\mathbb{R}$ which are unimodular and of type (R).

Keywords

left-invariant minimal unit vector field;Lie group;semi-direct product

References

  1. E. Boeckx and L. Vanhecke, Harmonic and minimal radial vector fields, Acta Math. Hungar. 90 (2001), no. 4, 317-331. https://doi.org/10.1023/A:1010687231629
  2. E. Boeckx and L. Vanhecke, Harmonic and minimal vector fields on tangent and unit tangent bundles, Differential Geom. Appl. 13 (2000), no. 1, 77-93. https://doi.org/10.1016/S0926-2245(00)00021-8
  3. O. Gil-Medriano and E. Llinares-Fuster, Minimal unit vector fields, Tohoku Math. J. (2) 54 (2002), no. 1, 71-84. https://doi.org/10.2748/tmj/1113247180
  4. O. Gil-Medriano and E. Llinares-Fuster, Second variation of volume and energy of vector fields. Stability of Hopf vector fields, Math. Ann. 320 (2001), no. 3, 531-545. https://doi.org/10.1007/PL00004485
  5. H. Gluck and W. Ziller, On the volume of a unit vector field on the three-sphere, Comment. Math. Helv. 61 (1986), no. 2, 177-192. https://doi.org/10.1007/BF02621910
  6. J. C. Gonzalez-Davila and L. Vanhecke, Examples of minimal unit vector fields, Special issue in memory of Alfred Gray (1939-1998), Ann. Global Anal. Geom. 18 (2000), no. 3-4, 385-404. https://doi.org/10.1023/A:1006788819180
  7. J. C. Gonzalez-Davila and L. Vanhecke, Minimal and harmonic characteristic vector fields on three-dimensional contact metric manifolds, J. Geom. 72 (2001), no. 1-2, 65-76. https://doi.org/10.1007/s00022-001-8570-4
  8. J. C. Gonzalez-Davila and L. Vanhecke, Energy and volume of unit vector fields on three-dimensional Riemannian manifolds, Differential Geom. Appl. 16 (2002), no. 3, 225-244. https://doi.org/10.1016/S0926-2245(02)00060-8
  9. D. L. Johnson, Volumes of flows, Proc. Amer. Math. Soc. 104 (1988), no. 3, 923-931. https://doi.org/10.1090/S0002-9939-1988-0964875-4
  10. O. Kowalski and Z. Vlasek, omogeneous Riemannian manifolds with only one homogeneous geodesic, Publ. Math. Debrecen 62 (2003), no. 3-4, 437-446.
  11. J. B. Lee, K. B. Lee, J. Shin, and S. Yi, Unimodular groups of type ${\mathbb{R}}^3{\rtimes}{\mathbb{R}}$, J. Korean Math. Soc. 44 (2007), no. 5, 1121-1137. https://doi.org/10.4134/JKMS.2007.44.5.1121
  12. J. Milnor, Curvatures of left invariant metrics on Lie groups, Advances in Math. 21 (1976), no. 3, 293-329. https://doi.org/10.1016/S0001-8708(76)80002-3
  13. S. L. Pedersen, Volumes of vector fields on spheres, Trans. Amer. Math. Soc. 336 (1993), no. 1, 69-78. https://doi.org/10.2307/2154338
  14. W. A. Poor, Differential Geometric Structures, McGraw-Hill Book Co., New York, 1981.
  15. M. Salvai, OOn the volume of unit vector fields on a compact semisimple Lie group, J. Lie Theory 13 (2003), no. 2, 457-464.
  16. K. Tsukada and L. Vanhecke, Invariant minimal unit vector fields on Lie groups, Period. Math. Hungar. 40 (2000), no. 2, 123-133. https://doi.org/10.1023/A:1010331408580
  17. S. Yi, Left-invariant minimal unit vector fields on a Lie group of constant negative sectional curvature, Bull. Korean Math. Soc. 46 (2009), no. 4, 713-720. https://doi.org/10.4134/BKMS.2009.46.4.713