  The purpose of this article is to characterize symplectic and Hamiltonian
circle actions on symplectic manifolds in terms of symplectic embeddings of
Riemann surfaces.
  More precisely, we will show that (1) if $(M,\omega)$ admits a Hamiltonian
$S^1$-action, then there exists an $S^1$-invariant symplectic $2$-sphere $S$ in
$(M,\omega)$ such that $\langle c_1(M), [S] \rangle > 0$, and (2) if the action
is non-Hamiltonian, then there exists an $S^1$-invariant symplectic
  $2$-torus $T$ in $(M,\omega)$ such that $\langle c_1(M), [T] \rangle = 0$.
  As applications, we will give a very simple proof of the following well-known
theorem which was proved by Atiyah-Bott \cite{AB}, Lupton-Oprea \cite{LO}, and
Ono \cite{O2} : suppose that $(M,\omega)$ is a smooth closed symplectic
manifold satisfying $c_1(TM)=\lambda \cdot [\omega]$ for some $\lambda \in \R$
and let $G$ be a compact connected Lie group acting effectively on $M$
preserving $\omega$. Then (1) if $\lambda < 0$, then $G$ must be trivial, (2)
if $\lambda=0$, then the $G$-action is non-Hamiltonian, and (3) if $\lambda >
0$, then the $G$-action is Hamiltonian.
