On Ricci curvatures of left invariant metrics on SU(2)
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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∈Χ(M) } where $Ric$ is the Ricci tensor field on (M,g), and then get a necessary and sufficient condition for the Levi-Civita connection ▽ 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.
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∈Χ(M) } where $Ric$ is the Ricci tensor field on (M,g), and then get a necessary and sufficient condition for the Levi-Civita connection ▽ 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.
참고문헌 (9)
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