An Isometrical C Pⁿ-Theorem

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Let Mⁿ(n≥3) be a complete Riemannian manifold with ₘ≥ 1, and let Mᵢⁿⁱ (i=1,2) be two comlplete totally geodesic submanifolds in M. We prove that if n₁+n₂=n-2 and if the distance |M₁M₂|≥π2, then Mᵢ is isometric to Sⁿⁱ/ Zₕ, C P nᵢ2 or C P nᵢ2/ Z₂ with the canonical metric when nᵢ>0, and thus M is isometric to Sⁿ/ Zₕ, C P n2 or C P n2/ Z₂ except possibly when n=3 and M₁ (or M₂) iso S¹/ Zₕ with h≥ 2 or n=4 and M₁ (or M₂) isoRP².