## 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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### References:

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