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In this paper, we extend the work of Cao-Chen [9] on Bach-flat gradient Riccisolitons to classify $n$-dimensional ($n\ge 5$) complete $D$-flat gradientsteady Ricci solitons. More precisely, we prove that any $n$-dimensionalcomplete noncompact gradient steady Ricci soliton with vanishing $D$-tensor iseither Ricci-flat, or isometric to the Bryant soliton. Furthermore, the proofextends to the shrinking case and the expanding case as well.

On Complete Gradient Steady Ricci Solitons with Vanishing D-tensor