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Our main aim in this paper is to investigate the rigidity of completenoncompact gradient steady Ricci solitons with harmonic Weyl tensor. Moreprecisely, we prove that an $n$-dimensional ($n\geq 5$) complete noncompactgradient steady Ricci soliton with harmonic Weyl tensor and multiply warpedproduct metric is either Ricci flat or isometric to the Bryant soliton up toscaling. Meanwhile, for $n\ge 5$, we provide a local structure theorem for$n$-dimensional connected (not necessarily complete) gradient Ricci solitonswith harmonic Weyl curvature and multiply warped product metric.

Rigidity of Complete Gradient Steady Ricci Solitons with Harmonic Weyl Curvature