# The rigidity on the second fundamental form of projective manifolds

## TL;DR

This paper derives a formula for the mean squared length of the second fundamental form of complex projective manifolds and classifies cases with small mean values, confirming conjectures and improving existing results.

## Contribution

It provides a new formula for the mean squared second fundamental form and classifies manifolds with small mean values, confirming conjectures and extending previous results.

## Key findings

- Confirmed a conjecture characterizing linear subspaces and quadrics.
- Improved a result from pointwise to mean case for a classical problem.
- Established an optimal second gap value for the mean squared second fundamental form.

## Abstract

Let $M$ be a complex $n$-dimensional projective manifold in $\mathbb{P}^{n+r}$ endowed with the Fubini-Study metric of constant holomorphic sectional curvature $1$, $\sigma$ its second fundamental form, and $\underline{|\sigma|}^2$ the mean value of the squared length of $\sigma$ on $M$. We derive a formula for $\underline{|\sigma|}^2$ and classify them when $\underline{|\sigma|}^2\leq2n$. We present several applications to these results. The first application is to confirm a conjecture of Loi and Zedda, which characterizes the linear subspace and the quadric in terms of the $L^2$-norm of $\sigma$. The second application is to improve a result of Cheng solving an old conjecture of Oguie from pointwise case to mean case. The third application is to give an optimal second gap value on $\underline{|\sigma|}^2$, which can be viewed as a complex analog to those on minimal submanifolds in the unit spheres.