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

다국어 초록 (Multilingual 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.

다국어 초록 (Multilingual 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 (Reference)

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2 J.-S. Park, "The Abbena-Thurston manifold as a critical point" 39 (39): 352-359, 1996

3 J.-S. Park, "Harmonic inner automorphisms of compact connected semisimple Lie groups" 42 (42): 83-91, 1990

4 S. Kobayashi, "Foundations of Differential Geometry" Wiley-Interscience 1963

5 A. L. Besse, "Einstein Manifolds" Springer-Verlag 1987

6 J. Milnor, "Curvatures of left invariant metrics on Lie groups" 21 (21): 293-329, 1976

7 J. A.Wolf, "Curvature in nilpotent Lie groups" 15 : 271-274, 1964

8 J.-S. Park, "Critical homogeneous metrics on the Heisenberg manifold" 11 (11): 31-34, 2005

9 K. Nomizu, "Affine Differential Geometry: Geometry of Affine Immersions" Cambridge University Press 1994