The Gauss map is assumed to be continuous. There does arise the question of choosing the normal vector among the two possible choices (namely and ). No jumps are allowed. There is no a priori "better" choice: being able to choose a continuous Gauss map is equivalent to the surface being orientable. That is always possible locally (i.e. on a small piece of the surface).
The notion of Gauss map can be generalized to a submanifold of dimension in an ambient manifold of dimension . In that case, the unit normal vector is replaced by the tangent -plane at the point . (It should be noted that in euclidean 3-space, a 2-plane is characterized by its unit normal vector -- up to sign, hence the definition above.) The Gauss map then goes from to the set of tangent -planes in the tangent bundle . The situation is somewhat simpler inside Euclidean space of dimension , where the target set is the oriented Grassmannian : the set of all oriented -planes in . (In the introductory example, .) In a more general manifold , the target space for the Gauss map is a Grassmann bundle built on the tangent bundle . The question of the global existence of is then a non-trivial topological question.