Question 1
Specifically, if we have an orientable $n$-dimensional manifold $\mathcal{M}$ embedded in $\mathbb{R}^{n+1}$ and we require that the class of all outward-facing normal vectors $\mathcal{N}_\mathcal{M}$ at all points of $\mathcal{M}$ form a vector space, does this force the manifold to be smooth or analytic? Being closed under vector addition seems like it would make the manifold smooth, and the existence of inverses makes me think that the surface would have to be closed in the $2$-manifold case.
EDIT: I believe this can be phrased also as: if the image of the Gauss map for an $n$-dimensional surface in $\mathbb{R}^{n+1}$ has no 'holes' in it, does this impose any particularly nice conditions on the surface? It seems like the Gauss-Bonnet theorem might come into play here?
Question 2
If we drop the requirement that inverses exist but maintain all the other structure requirements for a vector field, does this allow for open manifolds like a (finite) plane?
EDIT: I believe that this is equivalent to allowing for a non-symmetric image for the the surface under the Gauss map, but this might not be what I want to say. (is there a 'canonical' Gauss mapping?)
Does this change for non-orientable manifolds? Any references to appropriate literature would be greatly appreciated.