Hope this isn't a duplicate.
Let $S$ be an n-surface. Now since for any finite dimensional Topological space, connectedness and path-connectedness are equivalent notions, to show an n-surface to be path-connected we can proceed to show that it's connected and to show connectedness there are lots of tools. It tells that given any two points in $S$, $\exists$ a path joining those two points lying entirely within $S$.
But currently I'm trying to construct an explicit path between two points in $S$ . So all we have are arbitrary two points $x$ , $y$ $\in S$ and an equation of the form , say, $f(x_1 ,x_2,\ldots ,x_{n+1}) = c$ for some $f : {\Bbb R}^{n+1} \to \Bbb R$ , $f$ is smooth , $c \in \Bbb R$ is a regular value of $f$ (i.e. $f(x_1 ,x_2,\ldots ,x_{n+1}) = c$ is a level set) .
So, firstly I was looking to construct it for some relatively familiar surfaces like Annulus and then to chalk out a strategy for a general n-surface. But I can't figure out anything!
If someone can help by first giving an explicit path in a particular surface and then share some ideas on how to construct one such for a general n-surface , it will be extremely helpful .