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On the geometry of the Bochner tensor of nearly Kähler manifolds. (English. Russian original) Zbl 0818.53085

Russ. Math. Surv. 48, No. 5, 155-156 (1993); translation from Usp. Mat. Nauk 48, No. 5(293), 155-156 (1993).
The authors present some theorems on the geometry of the Bochner tensor of a nearly Kähler manifold whose fundamental form is a Killing form. These theorems generalize some results of S. Tachibana and R. C. Liu [Kodai Math. J. 22, 313-321 (1970; Zbl 0199.253)], M. Matsumoto [Tensor, New Ser. 20, 25-28 (1969; Zbl 0174.250)], M. Seino [Hokkaido Math. J. 11, 216-224 (1982; Zbl 0521.53053)] and other. For example:
Theorem 4. The HB-curvature of a nearly Kähler manifold is pointwise constant if and only if it is globally constant. A nearly Kähler manifold is a manifold of pointwise constant HB-curvature if and only if it is locally holomorphically isometric to one of the following manifolds: 1. \(\mathbb{C}^ n\); 2. \(\mathbb{C} P^ n\); 3. \(\mathbb{C} H^ n\); 4. \(S^ 6\); 5. \(M^ 2\); 6. \(\mathbb{C} P^ n(c) \times \mathbb{C} H^ m (-c)\); 7. \(\mathbb{C} P^ n (c)\times M^ 2 (-c)\); 8. \(\mathbb{C} H^ m (-c)\times M^ 2 (c)\); 9. \(M^ 2 (c)\times M^ 2 (-c)\) endowed with the canonical nearly Kähler structure, where \(c\) is the modulus of the holomorphic sectional curvature of the corresponding complex space form.
Reviewer: A.Neagu (Iaşi)

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds
32Q99 Complex manifolds
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