A new Lichnerowicz-Obata estimate in the presence of a parallel p-form

Let (M-n, g) be a compact Riemannian manifold with a smooth boundary. In this paper, we give a Lichnerowicz-Obata type lower bound for the first eigenvalue of the Laplacian of (M-n, g) when M has a parallel p-form (2 less than or equal to p less than or equal to n/2). This result follows from a new Bochner-Reilly's formula. Moreover, we give a characterization of the equality case when (M-n, g) is simply connected.