arXiv:2610.31720v1 Announce Type: new Abstract: In this paper, we first derive the Bochner-type formulas of the Hermitian-Tanno connection on contact Riemannian manifolds. Second, we prove a sub-Laplacian comparison theorem on complete noncompact contact Riemannian manifolds with bounded geometric conditions. Finally, we obtain the gradient estimates of a positive eigenfunction on a complete noncompact contact Riemannian manifold which satisfies similar geometric conditions.
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