I am studying Riemannian Geometry by myself and I am following the textbook Riemannian Geometry by Manfredo Perdigao do Carmo.
The Riemannian metric on a Lie group G is given by defining it on the tangent space of the neutral element e and it is given by:
$$\langle u,v\rangle_x = \langle (dL_{x^{-1}})_x(u),(dL_{x^{-1}})_x(v)\rangle_e\ ,\text{ where } x \in G, u,v \in T_xG.$$
I have verified all the properties of a Riemannian metric except for the smoothness.
Since a Riemannian metric is a 2-tensor and not a differential form, I am not sure how to understand it's pullback (by $L_{x^{-1}})$. Can somebody explain how the metric is varying smoothly due to the smoothness of $L_{x^{-1}}$. Thanks in advance.
Edit: I also want to know how to find an explicit formula for a Riemannian metric on $S^n$ which is induced by the pullback of the inclusion map from $\Bbb R^n$ to $S^n$.