Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2018, Director: Ignasi Mundet i Riera[en] The goal of this work is to prove a non existence theorem of non-trivial $S^{1}$ actions on a certain kind of smooth manifolds. More specifically, let $T$ be the $n$-dimensional torus and $M$ a smooth conected, closed (i.e. compact and without bondary) and orientable manifold of dimension $n$ such that $\chi(T \# M) \neq 0$. Then there are no non-trivial $S^{1}$ actions on $T \neq M$. Before proving this statement, some smooth manifold and Lie group theory will be developed: the proof of the Sard and the Poincaré-Hopf theorems stand out in this part.