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Characterizations of complex space forms by means of geodesic spheres and tubes

J. Gillard — 1996

Colloquium Mathematicae

We prove that a connected complex space form ( M n ,g,J) with n ≥ 4 can be characterized by the Ricci-semi-symmetry condition R ˜ X Y · ϱ ˜ = 0 and by the semi-parallel condition R ˜ X Y · σ = 0 , considering special choices of tangent vectors X , Y to small geodesic spheres or geodesic tubes (that is, tubes about geodesics), where R ˜ , ϱ ˜ and σ denote the Riemann curvature tensor, the corresponding Ricci tensor of type (0,2) and the second fundamental form of the spheres or tubes and where R ˜ X Y acts as a derivation.

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