Deformation of Sasakian metrics

Abstract

Deformations of the Reeb flow of a Sasakian manifold as transversely Kähler flows may not admit compatible Sasakian metrics anymore. We show that the triviality of the (0,2)-component of the basic Euler class characterizes the existence of compatible Sasakian metrics for given small deformations of the Reeb flow as transversely holomorphic Riemannian flows. We also prove a Kodaira-Akizuki-Nakano type vanishing theorem for basic Dolbeault cohomology of homologically orientable transversely Kähler foliations. As a consequence of these results, we show that any small deformations of the Reeb flow of a positive Sasakian manifold admit compatible Sasakian metrics.

35 pages, mistakes in lines 37--41 of page 12 of the previous version were corrected, a general revise was done to make the manuscript essentially self-contained

References in corpus (1)

• Sasakian Geometry, Hypersurface Singularities, and Einstein Metrics

Cited by in corpus (3)

• Rigidity of the Alvarez class
• Rigidity of the Álvarez classes of Riemannian foliations with nilpotent structure Lie algebras
• A note on the basic Lichnerowicz cohomology of transversally locally conformally Kählerian foliations