On Extensions of Myers' Theorem

Let M be a compact Riemannian manifold, and let h be a smooth function on M . Let p h ( x ) = inf|υ|− 1 (Ric x (υ,υ)−2Hess( h x (υ,υ)). Here Ric x denotes the Ricci curvature at x and Hess( h ) is the Hessian of h . Then M has finite fundamental group if Δ h − p h <0. Here Δ h =:Δ+2 L ∇ h is the Bismut‐Witten Laplacian. This leads to a quick proof of recent results on extension of Myers' theorem to manifolds with mostly positive curvature. There is also a similar result for noncompact manifolds.