does a manifold have to be embedded in a higher-dimensional space to be called a manifold? A 2-dimensional plane within a 3-dimensional space is a 2-dimensional manifold.
However, if we see the 2-dimensional plane simply as being embedded in a 2 dimensional space, and assume that there is no 3rd dimension at all, is this plane then still classified as a manifold?
And what if the 2-dimensional plane is unbounded in all directions, thereby being identical to the 2-dimensional space in which it is embedded, is it then still called a manifold?
(the definitions that I find suggest yes, but I'm not 100% sure).
 A: This is a funny instance where the plain English connotation of your question and the mathematical interpretation can differ.

*

*It is in fact true (Whitney's embedding theorem) that every manifold (for suitable definitions of manifold, some people count the long-line as a manifold) can be embedded in some higher dimensional Euclidean space.
So if we understand the question in your title to mean: is it true that every manifold is necessarily a submanifold of a higher dimensional space? The answer is yes.


*Furthermore, in some introductory textbooks of differential geometry of curves and surfaces, a manifold is in fact defined to be a submanifold of a higher dimensional space. This is generally a conscious choice made to make the subject more tangible for beginning students, and in view of the first point, is not really any loss in generality.


*However, it is true that most modern geometers and topologists consider manifolds devoid of an ambient embedding; that is at least in part because of the discovery (going back to Gauss and Riemann) that there are intrinsic quantities/qualities of manifolds that can be studied independently of any particular chosen embedding.
So if we interpret your question as asking whether "embedded as a submanifold of some higher dimensional space" is included as one of the defining conditions of a manifold, the answer is "frequently not".
