ABSTRACT. By means of a Kaluza–Klein type argument we show that the Perelman’sF–functional is the Einstein–Hilbert action in a space with extra “phantom ” dimensions. In this way, we try to interpret some remarks of Perelman in the introduction and at the end of the first section in his famous paper . As a consequence the Ricci flow (modified by a diffeomorphism and a time–dependent factor) is the evolution of the “real ” part of the metric under a constrained gradient flow of the Einstein–Hilber
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