 Embedding of the n-dimensional torus in codimension one

Suppose a compact $m$-manifold $N$ is embedded in the half-space $x_{m+1}>0$ of ${\mathbb{R}}^{m+1}$. Spinning it around the boundary hyperplane gives the embeddingStarting with a circle and iterating proves that $T^n=(S^1{)}^n$ embeds in ${\mathbb{R}}^{n+1}$.