  By means of a Kaluza-Klein type argument we show that the Perelman's
F-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 first 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-Hilbert gravitational action
in higher dimension.
