  Considering a n-dimensional general spacetime, we deduce its 4-dimensional
Einstein equation and Friedman equations, and discover a general dual relation
between the scale factor $a(t)$ of our universe and the scale factor $B(t)$ of
extra dimensions. Based on the dual relation equation, predictions of shrinking
of extra dimensions and free of singularity problem of our universe are given.
Therefore, solution to initial singularity paradox of our universe is achieved.
Because the dual relation is general, this Letter discovers that it is just the
extra dimensional shrinking contribution that results in our universe's
expanding in terms of the dual relation in the bulk space, and actually the
dual relation is deduced doesn't depend on the 4-dimensional matter concrete
Lagrangian, these are key important for a lot of future relative
investigations.
