A nullity condition for complex contact metric manifolds
A complex contact manifold is a complex manifold of odd complex dimension \(2n+1\) together with an open covering \(\{\mathcal{O}_{\alpha} \}\) by coordinate neighborhoods such that: (1) On each \(\mathcal{O}_{\alpha}\) there is a holomorphic 1-form \(\omega_{\alpha}\) such that \(\omega_{\alpha}\wedge (d\omega_{\alpha})^n\neq 0\); (2) On \(\mathcal{O}_{\alpha}\cap \mathcal{O}_{\beta}\neq \emptyset\) there is a non-vanishing holomorphic function \(\lambda_{\alpha\beta}\) such that \(\omega_{\alpha}= \lambda_{\alpha\beta} \omega_{\beta}\). The subspaces \(\{X \in T_m{\mathcal{O}}_{\alpha}: \omega_{\alpha}(X)= 0 \}\) define a non-integrable holomorphic subbundle \(\mathcal{H}\) of complex dimension \(2n\) called the complex contact subbundle or horizontal subbundle. A complex contact manifold \(M\) admits a complex almost contact metric structure, i.e. local real 1-forms \(u\) and \(v=uJ\), \((1,1)\) tensor fields \(G\) and \(H=GJ\) and a Hermitian metric \(g\) such that \(H^2= G^2 = -\text{Id} + u\otimes U + v\otimes V\), \(GJ=-JG\), \(GU=0\), \(g(X,GY)= -g(GX,Y)\), where \(U\) and \(V=-JU\) are the orthogonal dual vector fields to \(u\) and \(v\), respectively. On the other hand, if \((M,\varphi, \xi, \eta, g)\) is a contact metric manifold we say that \(\xi\) belongs to the \((\kappa,\mu)\)-nullity distribution if \(R_{XY}\xi= \kappa(\eta(Y)X-\eta(X)Y) + \mu(\eta(Y)hX-\eta(X)hY)\), where \(h= \frac{1}{2}\mathcal{L}_{\xi}\varphi\) and \(\kappa, \mu\) are constants. This nullity condition was studied by \textit{D. E. Blair}, \textit{T. Koufogiorgos} and \textit{B. J. Papantoniou} [Isr. J. Math. 91, No. 1--3, 189--214 (1995; Zbl 0837.53038)]. They proved that the \((\kappa, \mu)\)-nullity condition determines the curvature of \(M\) completely. Later, \textit{E. Boeckx} [Ill. J. Math 44, No. 1, 212--219 (2000; Zbl 0969.53019)] gave a full local classification by explicit examples. In this paper, the author introduces a nullity condition for complex contact metric manifolds, studies some curvature properties and proves that the curvature tensor is completely determined on such manifolds.
scientific article; zbMATH DE number 1982547