Find an example of n-dimensional differentiable manifold

Question

Find an example of $n$-dimensional differentiable manifold whose points are not points of the variety $\mathbb{R}^{n}$

Answer 1

$\mathbb{RP}^n$, the space of lines through the origin in $\mathbb R^{n+1}$, is not a real affine variety and is $n$-dimensional.