On Ricci curvatures of left invariant metrics on $SU(2)$
Bull. Korean Math. Soc. 2009 Vol. 46, No. 2, 255-261
https://doi.org/10.4134/BKMS.2009.46.2.255
Printed March 1, 2009
Yong-Soo Pyo, Hyun Woong Kim, and Joon-Sik Park
Pukyong National University, Pukyong National University, and Pusan University of Foreign Studies
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 $\{ r(X) := Ric(X,X) \ | \ ||X||_g =1, X \in {\frak X}(M) \}$, 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 $X$.
Keywords : Ricci curvature, left invariant metric, projectively flat
MSC numbers : 53C07, 53C25
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