Smooth Manifold

Definition

A smooth manifold is an infinitely Differentiable Manifold ($C^{\infty }$).

Facts

The Product Space of an $n$-dimensional smooth manifold and an $m$-dimensional smooth manifold is an $(n+m)$-dimensional smooth manifold.

Consider an $n$-dimensional smooth manifold without boundary $X$ and an $m$-dimensional smooth manifold with boundary $Y$. Then, $\partial (X\times Y)=X\times \partial Y$