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The structure of some classes of \(K\)-contact manifolds. (English) Zbl 1155.53020

Let \((M,g)\) be a Riemannian manifold of dimension \(m\geq 3\). Its projective curvature tensor is \( \mathcal {P}(X,Y)Z =R(X,Y)Z - \frac{1}{m-1}[g(Y,Z)QX- g(Y,Z)QY]\) , where \(R\) is the curvature tensor and \(Q\) is the Ricci operator. \((M,g)\) is projectively flat i.e. \(\mathcal {P}= 0\) if and only if \((M,g)\) is of constant curvature. Let now M be endowed with an almost contact metric structure \((\phi,\xi,\eta,g); m = 2n + 1\). The projective curvature tensor is the same as above. An almost contact metric manifold M is said to be:
– quasi projectively flat if \(g (\mathcal {P}(X,Y)Z,\phi W)) = 0, X,Y,Z,W \in TM\)
– \(\xi \)-projectively flat if \(\mathcal {P}(X,Y)\xi = 0,\)
– \(\phi \) -projectively flat if \(g(\mathcal {P}(\phi X,\phi Y)\phi Z, \phi W) = 0\).
– a \(K\) -contact manifold if \(\nabla \xi =- \phi,\) where \(\nabla \) is the Levi-Civita connection. The authors prove:
Theorem 3.3. If a \(K\)-contact manifold is quasi projectively flat then it is Einstein.
Theorem 3.5. Let \(M\) be a \((2n+1)\) - dimensional Sasakian manifold. Then the following statements are equivalent: (a) \(M\) is quasi projectively flat, (b) \(M\) is \(\phi \)-projectively flat (c) \(M \) is locally isometric to the unit sphere \(S^{2n+1}(1)\).
Theorem 4.1. A \(\phi \)-projectively flat compact regular \(K\)-contact manifold is a principal \(S^1\)-bundle over an almost Kähler space of constant holomorphic sectional curvature 4.

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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[1] Blair, D. E., Riemannian geometry of contact and symplectic manifolds (2002), Boston, MA: Birkhauser Boston, Inc., Boston, MA · Zbl 1011.53001
[2] Boothby, W. M.; Wang, H. C., On contact manifolds, Ann. Math., 68, 721-734 (1958) · Zbl 0084.39204
[3] Cabrerizo, J. L.; Fernández, L. M.; Fernández, M.; Zhen, G., The structure of a class of K-contact manifolds, Acta Math. Hungar., 82, 4, 331-340 (1999) · Zbl 0924.53024
[4] Mishra, R. S., On Sasakian manifolds, Indian J. Pure Appl. Math., 1, 98-105 (1970) · Zbl 0207.20504
[5] Ogiue, K., On fibrings of almost contact manifolds, Kodai Math. Sem. Rep., 17, 53-62 (1965) · Zbl 0136.18101
[6] Ozgur, C., φ-conformally flat Lorentzian para-Sasakian manifolds, Rad. Mat., 12, 1, 99-106 (2003) · Zbl 1074.53057
[7] Yano K and Bochner S, Curvature and Betti numbers, Annals of Mathematics Studies 32 (Princeton University Press) (1953)
[8] Zhen, G., On conformal symmetric K-contact manifolds, Chinese Quart. J. Math., 7, 5-10 (1992) · Zbl 0963.53051
[9] Zhen, G.; Cabrerizo, J. L.; Fernández, L. M.; Fernández, M., On ξ-conformally flat contact metric manifolds, Indian J. Pure Appl. Math., 28, 725-734 (1997) · Zbl 0882.53031
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