대한수학회지 (Journal of the Korean Mathematical Society)

  • 제60권1호
  • /
  • Pages.143-165
  • /
  • 2023
  • /
  • 0304-9914(pISSN)
  • /
  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

LEFT INVARIANT LORENTZIAN METRICS AND CURVATURES ON NON-UNIMODULAR LIE GROUPS OF DIMENSION THREE

  • Ku Yong Ha (Department of mathematics Sogang University) ;
  • Jong Bum Lee (Department of mathematics Sogang University)
  • 투고 : 2022.05.17
  • 심사 : 2022.10.27
  • 발행 : 2023.01.01
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

For each connected and simply connected three-dimensional non-unimodular Lie group, we classify the left invariant Lorentzian metrics up to automorphism, and study the extent to which curvature can be altered by a change of metric. Thereby we obtain the Ricci operator, the scalar curvature, and the sectional curvatures as functions of left invariant Lorentzian metrics on each of these groups. Our study is a continuation and extension of the previous studies done in [3] for Riemannian metrics and in [1] for Lorentzian metrics on unimodular Lie groups.

키워드

과제정보

The authors would like to thank Professor Kyung Bai Lee for thorough reading and valuable comments in their original version. These comments made our exposition much more transparent than before.

참고문헌

  1. M. Boucetta and A. Chakkar, The moduli spaces of Lorentzian left-invariant metrics on three-dimensional unimodular simply connected Lie groups, J. Korean Math. Soc. 59 (2022), no. 4, 651-684. https://doi.org/10.4134/JKMS.j210460 
  2. G. Calvaruso, Homogeneous structures on three-dimensional Lorentzian manifolds, J. Geom. Phys. 57 (2007), no. 4, 1279-1291. https://doi.org/10.1016/j.geomphys.2006.10.005 
  3. K. Y. Ha and J. B. Lee, Left invariant metrics and curvatures on simply connected three-dimensional Lie groups, Math. Nachr. 282 (2009), no. 6, 868-898. https://doi.org/10.1002/mana.200610777 
  4. K. Y. Ha and J. B. Lee, Left invariant Lorentzian metrics and curvatures on non-unimodular Lie groups of dimension three, arXiv:2209.02208. 
  5. 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 
  6. K. Nomizu, Left-invariant Lorentz metrics on Lie groups, Osaka Math. J. 16 (1979), no. 1, 143-150. http://projecteuclid.org/euclid.ojm/1200771834 
  7. B. O'Neill, Semi-Riemannian Geometry, Academic Press, Inc., New York, 1983. 
  8. S. V. Thuong, Metrics on 4-dimensional unimodular Lie groups, Ann. Global Anal. Geom. 51 (2017), no. 2, 109-128. https://doi.org/10.1007/s10455-016-9527-z