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

## TL;DR

This paper establishes a Lichnerowicz-Obata type estimate for the first Laplacian eigenvalue on manifolds with an almost parallel p-form and explores manifold decomposition under certain pinching conditions.

## Contribution

It introduces a new eigenvalue estimate for manifolds with almost parallel p-forms and provides an almost decomposition result under specific geometric pinching conditions.

## Key findings

- Eigenvalue estimate for manifolds with almost parallel p-forms
- Almost decomposition results under pinching conditions
- Extension of classical estimates to new geometric contexts

## 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 ($2\leq p \leq n/2$) in $L^2$-sense, and give an almost decomposition result of the manifold under some pinching conditions when $2\leq p<n/2$.