A note on Canonical Ricci forms on 2-step nilmanifolds

In this note we prove that any left-invariant almost Hermitian structure on a 2-step nilmanifold is Ricci-flat with respect to the Chern connection and that it is Ricci-flat with respect to another canonical connection if and only if it is cosymplectic.Comment: 7 pages. Typos corrected. To appear in Proc. of the AM

A generalization of the normal holomorphic frames in Symplectic Manifolds

In this paper we give a generalization of the normal holomorphic frames in the symplectic manifolds and find conditions for the integrability of complex structures.Comment: 10 page

Contact Calabi-Yau manifolds and Special Legendrian submanifolds

We consider a generalization of Calabi-Yau structures in the context of $\alpha$-Sasakian manifolds. We study deformations of a special class of Legendrian submanifolds and classify invariant contact Calabi-Yau structures on 5-dimensional nilmanifolds. Finally we generalize to codimension $r$.Comment: 16 pages, no figures. Final version to appear in "Osaka J. Math.

A remark on the Laplacian flow and the modified Laplacian co-flow in G2-Geometry

We observe that the DeTurck Laplacian flow of G2-structures introduced by Bryant and Xu as a gauge fixing of the Laplacian flow can be regarded as a flow of G2-structures (not necessarily closed) which fits in the general framework introduced by Hamilton in [4].Comment: 3 pages, comments are welcom