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

Title & Authors
ON RICCI CURVATURES OF LEFT INVARIANT METRICS ON SU(2)
Pyo, Yong-Soo; Kim, Hyun-Woong; Park, Joon-Sik;

Abstract
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) | $\small{{||X||}_g}$ = 1,X $\small{{\in}}$ 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 $\small{{\nabla}}$ 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.
Keywords
Ricci curvature;left invariant metric;projectively flat;
Language
English
Cited by
1.
YANG-MILLS CONNECTIONS ON CLOSED LIE GROUPS,;;;

호남수학학술지, 2010. vol.32. 4, pp.651-661
1.
YANG-MILLS CONNECTIONS ON CLOSED LIE GROUPS, Honam Mathematical Journal, 2010, 32, 4, 651
References
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

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

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