If we need to measure arc length, the rigorous way is as the following:

The part in the middle is tensor metric. but it is from point of extrinsic view, what if we were an ant on the surface made up of u and v, how to calculate the arc length? it’s similar as below:

Using this intrinsic approach and metric tensor, it’s also computing the arc length same result as using the extrinsic approach. In curved space, what you see with eyes are misguiding but with the aid of metric tensor, measuring length is also easy to do.