  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.
