A new Lichnerowicz-Obata estimate in the presence of a parallel \(p\)-form (Q697386)

Let \(\lambda_{1}\) denote the lowest positive eigenvalue of the Laplacian of a closed Riemannian manifold \(M^{n}\) with Ricci curvature \(\text{Ric}^{M}\geq kg\) for some \(k>0\), where \(g\) is the riemannian metric. Then we have the Lichnerowicz inequality \[ \lambda_{1}\geq\left(1+ {1 \over n-1}\right)k , \] and Obata's theorem states that the equality holds if and only if \(M\) is isometric to the euclidean \(n\)-dimensional sphere of radius \(\sqrt{k/(n-1)}\). Several sharpenings of these results were obtained when \(M\) can be endowed with other structures, and the paper contains a new theorem of this type. First, if \(M\) admits a nontrivial parallel \(p\)-form, \(2\leq p\leq n/2\), then the author shows that \[ \lambda_{1}\geq\left(1+ {1\over n-p-1}\right)k . \] An example of such a Riemannian manifold \(M\) where the equality holds is the product of the \((n-p)\)-dimensional Euclidean sphere of radius \(\sqrt{(n-p-1)/k}\) and a closed oriented \(p\)-dimensional Riemannian manifold \(Q\) with \(\text{Ric}^{Q}\geq kg\); in this case, a nontrivial parallel \(p\)-form on \(M\) is defined by the volume form of \(Q\). Conversely, if \(p<n/2\), \(M\) is simply connected and the equality holds, then \(M\) is a product as above. Analogous statements are also shown for manifolds with convex boundary under Dirichlet conditions. The proof uses a new Bochner-Reilly formula.