대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제46권2호
  • /
  • Pages.255-261
  • /
  • 2009
  • /
  • 1015-8634(pISSN)
  • /
  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

ON RICCI CURVATURES OF LEFT INVARIANT METRICS ON SU(2)

  • Pyo, Yong-Soo (DIVISION OF MATHEMATICAL SCIENCES PUKYONG NATIONAL UNIVERSITY) ;
  • Kim, Hyun-Woong (DEPARTMENT OF MATHEMATICS PUKYONG NATIONAL UNIVERSITY) ;
  • Park, Joon-Sik (DEPARTMENT OF MATHEMATICS PUSAN UNIVERSITY OF FOREIGN STUDIES)
  • 발행 : 2009.03.31
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

In this paper, we shall prove several results concerning Ricci curvature of a Riemannian manifold (M, g) := (SU(2), g) with an arbitrary given left invariant metric g. First of all, we obtain the maximum (resp. minimum) of {r(X) := Ric(X,X) | ${||X||}_g$ = 1,X ${\in}$ X(M)}, where Ric is the Ricci tensor field on (M, g), and then get a necessary and sufficient condition for the Levi-Civita connection ${\nabla}$ on the manifold (M, g) to be projectively flat. Furthermore, we obtain a necessary and sufficient condition for the Ricci curvature r(X) to be always positive (resp. negative), independently of the choice of unit vector field X.

키워드

참고문헌

  1. A. L. Besse, Einstein Manifolds, Springer-Verlag, Berlin, 1987
  2. S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. I, Wiley-Interscience, New York, 1963
  3. 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
  4. K. Nomizu and T. Sasaki, Affine Differential Geometry: Geometry of Affine Immersions, Cambridge University Press, Cambridge, 1994
  5. J.-S. Park, Harmonic inner automorphisms of compact connected semisimple Lie groups, Tohoku Math. J. (2) 42 (1990), no. 1, 83–91 https://doi.org/10.2748/tmj/1178227695
  6. J.-S. Park, Critical homogeneous metrics on the Heisenberg manifold, Interdiscip. Inform. Sci. 11 (2005), no. 1, 31–34
  7. J.-S. Park and W. T. Oh, The Abbena-Thurston manifold as a critical point, Canad. Math. Bull. 39 (1996), no. 3, 352–359
  8. K. Sugahara, The sectional curvature and the diameter estimate for the left invariant metrics on SU(2, C) and SO(3, R), Math. Japon. 26 (1981), no. 2, 153–159
  9. J. A.Wolf, Curvature in nilpotent Lie groups, Proc. Amer. Math. Soc. 15 (1964), 271–274

피인용 문헌

  1. YANG-MILLS CONNECTIONS ON CLOSED LIE GROUPS vol.32, pp.4, 2010, https://doi.org/10.5831/HMJ.2010.32.4.651