I am reading the beginning of the first volume on the introduction to Differential Geometry by Spivak.
In the first part he defines manifolds and states that 'the only connected 1-manifolds are the line $\mathbb{R}$ and the circle $\mathbb{S}^1$'.
(1) Is that 'only' a direct consequence of the definition of manifold? I do not really see the answer. It is also true that, at this point of the book, Spivak uses results from Algebraic topology/Homology (without proving them) to give a wider outlook on the world of manifolds. So it might well be that the answer is not trivial. If it actually is, I would like to see why.
(2) Does this fact imply that the only surfaces of revolution in $\mathbb{R}^3$ which can be obtained by connected 1-manifolds are cylinders and 2-tori?
You could try the following resources: this or this or (this (all lecture notes for classes). Or on this site as well, with bonus link. This one looks pretty formal.