Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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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)
  • Published : 2009.03.31
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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 ${\in}$ 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.

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References

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