We prove that a steady gradient Ricci soliton is either Ricci flat with a constant potential function or a quotient of the product steady soliton \(N^{n-1}\times \mathbb {R}\) , where \(N^{n-1}\) is Ricci flat, or isometric to the Bryant soliton (up to scalings), provided that a couple of geometric conditions inspired by the cigar soliton hold. As an application, we prove that any complete non-compact steady Ricci soliton with positive Ricci curvature controlled by the scalar curvature R, curvature tensor Rm satisfying \(|Rm|r\rightarrow o(1)\) and \(R\rightarrow \infty \) , as \(r\rightarrow \infty \) , must be the Bryant soliton. Moreover, we prove that any complete steady soliton with positively pinched Ricci curvature must be Ricci flat.
