  We explore compactifications of the form of three tori with a general genus
and one circle in the framework of 11D supergravity. By imposing suitable gauge
conditions and boundary conditions, we find that the FRW universe with four
extended spacetime dimensions and seven extremely small compactified spatial
dimensions emerges as a solution for the 11D supergravity. These specific
compactification methods can produce cosmological inflation that aligns with
the observational constraints set by the 2021 BICEP/Keck and Planck 2018
results. In the cosmological inflation models we construct, the inflaton can be
interpreted as the conformal vibrations of extra dimensions with a size around
10^5 times the reduced Planck length. Additionally, we offer the theoretical
predictions for the mass of the inflaton, and the tree-level Newton's gravity
law between two massive point particles surrounded by a spherically symmetric
distribution of the inflaton, which can reproduce the Tully-Fisher relation and
explain the flat rotation curves of galaxies.
