  We study the effective spacetimes in lower dimensions that can be extracted
from a multidimensional generalization of the Schwarzschild-Tangherlini
spacetimes derived by Fadeev, Ivashchuk and Melnikov ({\it Phys. Lett,} {\bf A
161} (1991) 98). The higher-dimensional spacetime has $D = (4 + n + m)$
dimensions, where $n$ and $m$ are the number of "internal" and "external" extra
dimensions, respectively. We analyze the effective $(4 + n)$ spacetime obtained
after dimensional reduction of the $m$ external dimensions. We find that when
the $m$ extra dimensions are compact (i) the physics in lower dimensions is
independent of $m$ and the character of the singularities in higher dimensions,
and (ii) the total gravitational mass $M$ of the effective matter distribution
is less than the Schwarzshild mass. In contrast, when the $m$ extra dimensions
are large this is not so; the physics in $(4 + n)$ does explicitly depend on
$m$, as well as on the nature of the singularities in high dimensions, and the
mass of the effective matter distribution (with the exception of wormhole-like
distributions) is bigger than the Schwarzshild mass. These results may be
relevant to observations for an experimental/observational test of the theory.
