The structure of some classes of \(K\)-contact manifolds (Q949459)

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.