Lichnerowicz-Obata Estimate, Almost Parallel $p$-form and Almost Product Manifolds

Abstract

We show a Lichnerowicz-Obata type estimate for the first eigenvalue of the Laplacian of $n$-dimensional closed Riemannian manifolds with an almost parallel $p$-form ($2\le p\le n/2$) in $L^2$-sense, and give an almost decomposition result of the manifold under some pinching conditions when $2\le p<n/2$.

66 pages v2:some error corrected, Appendix B added, v3:some error corrected, v4:some proofs simplified, v5: some error corrected, some descriptions simplified