  Kaluza-Klein approach in an N(=1+3+D)-dimensional Friedmann-Robertson-Walker
type space is often adopted in the literature. We derive a compact expression
for the Friedmann equation in a (1+3+D)-dimensional space. The redundancy of
the associated field equations due to the Bianchi identity is analyzed. We also
study the dilaton gravity theory with higher-derivative gravitational
couplings. It turns out that higher-order terms will not affect the Friedmann
equation in a constant flat internal space. This is true only for the flat-De
Sitter external space. The inflationary solution in an induced-gravity model is
also discussed as an application.
