Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2009; 46(2): 255-261

Printed March 1, 2009

https://doi.org/10.4134/BKMS.2009.46.2.255

Copyright © The Korean Mathematical Society.

On Ricci curvatures of left invariant metrics on $SU(2)$

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