Note on the conjecture of D.Blair in contact Riemannian geometry

TL;DR

This paper investigates D.Blair's conjecture in contact Riemannian geometry, proving it for specific cases on closed 3-manifolds, providing a local counterexample, and showing hyperbolic metrics on R^3 cannot be contact-compatible.

Contribution

It proves D.Blair's conjecture for certain contact structures on closed 3-manifolds, constructs a local counterexample, and demonstrates hyperbolic metrics on R^3 are incompatible with contact structures.

Findings

Proved D.Blair's conjecture for specific contact structures on closed 3-manifolds.

Constructed a local counterexample to the conjecture.

Showed hyperbolic metrics on R^3 cannot be compatible with any contact structure.

Abstract

The conjecture of D.Blair says that there are no nonflat Riemannian metrics of nonpositive curvature compatible with a contact structure. We prove this conjecture for a certain class of contact structures on closed 3-dimensional manifolds and construct a local counterexample. We also prove that a hyperbolic metric on $\mathbb{R}^3$ cannot be compatible with any contact structure.