In this talk, we will discuss the rigidity problem of steady  gradient Ricci solitons with positive curvature, in particular in  4 dimension.   As we know, there are two typical classes of steady  gradient Ricci solitons with positive curvature in higher dimension $d\ge 4$, namely,  one called as Bryant  Ricci solitons, another called as flying wings, recently constructed by Yi Lei. Our purpose is to give a character to  those two classes of  Ricci solitons based on  types  of   sequences of blow-down Ricci soliton flows.

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