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We apply the Berglund-Hübsch transpose rule from BHK mirror symmetry to show that to an $n-1$-dimensional Calabi-Yau orbifold in weighted projective space defined by an invertible polynomial, we can associate four (possibly) distinct Sasaki manifolds of dimension $2n+1$ which are $n-1$-connected and admit a metric of positive Ricci curvature. We apply this theorem to show that for a given K3 orbifold, there exists four seven dimensional Sasakian manifolds of positive Ricci curvature, two of which are actually Sasaki-Einstein., Fixed some typos and added references

In this paper we define the notion of a generalized coK��hler structure and prove that the product $M_{1}\times M_{2}$ of generalized contact metric manifolds $(M_i, ��_i,E_{\pm,i}, G_i)$, $ i=1, 2$, where $M_{1}\times M_{2}$ is endowed with the product generalized complex structure induced from $��_1$ and $��_2$, is generalized K��hler if and only if $(M_i, ��_i, E_{\pm,i}, G_i) ,\ \ i=1,2$ are generalized coK��hler structures. We also prove that products of generalized coK��hler and generalized K��hler manifolds admit a generalized coK��hler structure. We use these product constructions to give nontrivial examples of generalized coK��hler structures. Finally, we show the analogs of these theorems hold in the setting of twisted generalized geometries. We use these theorems to construct new examples of twisted generalized K��hler structures on manifolds that do not admit a classical K��hler structure and we give examples of twisted generalized coK��hler structures on manifolds which do not admit a classical coK��hler structure., Final print version. To appear in Journal of Geometry and Physics

Negative Sasakian manifolds, where the first Chern class of the contact subbundle is a torsion class, can be viewed as Seifert-$S^1$ bundles where the base orbifold has an ample orbifold canonical class. We use this framework to settle completely an open problem formulated by C.Boyer and K.Galicki which asks whether or not arbitrary connected sums of $S^2\times S^3$ are negative Sasakian manifolds. As a consequence of the affirmative answer to this problem, there exists so-called Sasaki eta-Einstein and Lorentzian Sasaki-Einstein metrics on all of these five-manifolds and moreover all of these can be realized as links of isolated hypersurface singularities defined by weighted homogenous polynomials. The key step is to construct infinitely many hypersurfaces with branch divisors in weighted projective that contain only rational curves., Comment: 12 pages. Corrected Corollary 4.4 and changed a few typos preceding this corollary