# differential geometry – Gauss map from an \$n\$ -sufrace is onto.

I was reading differential geometry from a lecture note, there I found the following theorem,

If $$S$$ is a compact connected oriented $$n-$$ surface in $$mathbb{R}^{n+1}.$$ Then the gauss map is surjective.

Now in the proof , the picture of the surface $$S$$ sketched as if $$S$$ is homeomorphic to $$S^n.$$ I am wondering that, is it true?

Is there any characterisation of $$n-$$ surcaces?

Definition: An $$n-$$ surface $$S$$ is a level curve of a smooth function whose gradient is non-zero on $$S.$$