2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/140612We construct a class of monotonic quantities along the normalized Ricci flow on closed n-dimensional manifolds.The only revision from the previous version dated October 23 1007 (yesterday) is that we have added a remark (Line 6 to line 4 from the bottom on Page 1) on our Referece [6] to make our citation clearerDifferential GeometryAnalysis of PDEsPrimary 53C21, 53C44; Secondary 58J35, 35P99A Class of Monotonic Quantities along the Ricci Flowtext