  Let $u_n$ be a sequence of mappings from a closed Riemannian surface $M$ to a
general Riemannian manifold $N$. If $u_n$ satisfies \beno \sup_{n}\big(\|\nabla
u_n\|_{L^2(M)}+\|\tau(u_n)\|_{L^{p}(M)}\big)\leq \Lambda\quad \text{for
some}\,\,p>1, \eeno where $\tau(u_n)$ is the tension field of $u_n$, then there
hold the so called energy identity and neckless property during blowing up.
This result is sharp by Parker's example, where the tension fields of the
mappings from Riemannian surface are bounded in $L^1(M)$ but the energy
identity fails.
