Comparison theorems in pseudo-Hermitian geometry and applications (Q1750371)

The authors consider pseudo-Hermitian manifolds \((M, H(M),J,\theta)\), where \(M\) is an \((2m+1)\)-dimensional differentiable manifold, \(H(M)\) is a \(2m\)-dimensional distribution on \(M\) endowed with a formally integrable complex structure \(J\), \(\theta\) is a nowhere vanishing one-form such that its Levi form \(L_{\theta}\) is positive definite. On such a manifold one introduces a special Riemannian metric \(g_{\theta}\) (the Webster metric) and a special connection with torsion (the Tanaka-Webster connection) \(\nabla\), which is better for the study of this structure than the Levi-Civita connection \(\nabla^{\theta}\). The authors build a theory of geodesics (Jacobi fields, relationship with the curvature, completeness, etc) for \(\nabla\) and obtain several results similar to the classical theorems of Hopf-Rinow, Cartan-Hadamard and Bonnet-Myers. For many of these results, the manifold is supposed to be also Sasakian, which means that a special one-form (obtained from the torsion of \(\nabla\) and a Reeb-type vector field) vanishes.