Conformally Sasakian manifolds and the $k$-nullity condition

arXiv:2611.12924v1 Announce Type: new Abstract: We study almost contact metric manifolds that are locally conformal to Sasakian manifolds in the normalization of Olszak. We show that the local conformal factors are uniquely determined by the structure; consequently, every such manifold is globally conformal to a Sasakian manifold and its Lee form is exact. We then investigate the $k$-nullity condition $ R(X,Y)\xi=k\{\eta(Y)X-\eta(X)Y\}. $ We prove that a smooth nullity function is necessarily constant, including in dimension three. Moreover, the $k$-nullity condition is equivalent to the assertion that $f=\exp(\sigma)$ satisfies $ \mathrm{Hess}_g f=-(kf+C)g $ for a constant $C$, and hence to the fact that the conformal metrics $g$ and $g_0=\exp(-2\sigma)g$ have the same geodesic circles. We derive the range of the conformal factor, a local warped-product description, and a classification in the complete case: when the Lee form is nonzero, the complete metric is a simply connected space form of curvature $k$. Finally, we construct incomplete examples of nonconstant sectional curvature in every dimension $2n+1\geq 5$.

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