surface vs differentiable manifold

Question

Every surface is a smooth manifold, but the reciprocal is verified?

some concrete example of a differentiable manifold is not surface?

Thanks in advance for the suggestions.

Answer 1

A surface is simply a two-dimensional manifold. However, some subtlety arises if you distinguish "topological manifolds" (Hausdorff, second-countable, locally Euclidean topological spaces), "differentiable manifolds" (with additional global differential structure), and "smooth manifolds" (where the structure is smooth, i.e. infinitely differentiable). See, for example, this.

Usually a surface is defined as a two-dimensional topological manifold; the answer might depend on context, in particular your definitions of "surface" and "smooth manifold".

Answer 2

The term surface is usually used to indicate a $2$-dimensional manifold (or even a $2$-dimensional submanifold of $\mathbb{R}^3$). Any manifold of higher or lower dimension would not be a surface.

More generally, the term hypersurface of $\mathbb{R}^n$ is used to denote a $(n-1)$-submanifold of $\mathbb{R}^n$.