On Curvatures of k-Adapted Metrics to Contact 3-Manifolds

Though there are many differential geometric results on contact metric structures, adapted metrics for contact structures introduced by S. Chern and R. Hamilton are not studied. In this paper, we study curvatures arising from k-adapted metric to contact structures on 3-manifolds, which is an extended notion of adapted metrics introduced by S. Chern and R. Hamilton, and give some results that also hold for contact metric structures, especially, on K-contact metric structures. We also study the Gaussian curvature of ξ introduced by V. Krouglov.