Tangent spaces and $\mathbb{R}^n$ The tangent space of a circle is a line.
The tangent space of a sphere (in every point) can be thought of as a plane. 
Is this a general thing? I mean, having an $n$ dimensional Riemannian manifold, can the tangent space in every point be thought as $\mathbb{R}^n$? 
If the answer is yes, does this happen as well with the Lorentzian manifolds of GR? Can the tangent space of any space-time always be regarded as a Minkowski space?

The answer is yes, the tangent space at any point of an $n$-dimensional real manifold is isomorphic to $\mathbb{R}^n$. What distinguishes the Riemannian from the Lorentzian case is that in the Riemannian case this vector space comes equipped with a positive definite metric, which can be rescaled so to give the use Euclidean metric, while in the Lorentzian case the tangent space comes equipped with a Lorentzian metric, which can b rescaled to give the Minkowskian metric in $n$ dimensions: diagonal with all pluses and one minus (or the opposite depending on your conventions).
