Norm riemann geometry, definition confused

Question

I am currently studying Riemannian geometry and defining the Riemannian metric as being given by $g(u,v)=u^iv^ig_{ij}$, but the norm is defined as $||u||=\sqrt{g(u,u)} =\sqrt{g_{ij}u^iu^j}$. Why in the definition of norm do we not have the same index $i=j$? since we are calculating $g(u,u)$ then we can write $u=u^i\partial_i$ and therefore we would have $g(u,u)=g_{ii}(u^i)^2$

Answer 1

The metric is a bilinear form with matrix $G=(g_{ij})$ that varies with the point, and may not be diagonal. The norm of a tangent vector is then $\|u\|^2=u^TGu=g_{ij}u^iu^j$. Only when $G$ is diagonal different coordinates of $u$ do not interact.