Splitting theorems in presence of an irrotational vector field

New splitting theorems in a semi-Riemannian manifold which admits an irrotational vector field (not necessarily a gradient) with some suitable properties are obtained. According to the extras hypothesis assumed on the vector field, we can get twisted, warped or direct decompositions. Some applications to Lorentzian manifold are shown and also ${\mathbf{S}}^1\times L$ type decomposition is treated.