From the coordinate-independent point of view, a metric tensor field is defined to be a nondegenerate symmetric bilinear form on each tangent space that varies smoothly from point to point.
I'm not sure I truly get this. The metric is locally a flat Lorentzian metric. We can write it in a variety of coordinates (and use a standard measure stick). But if a manifold is curved we have to include variations of the metric to say if the manifold is curved. But if it's locally flat how then can it encode for curvature (if it contains this information in the first place, of which I'm not sure)?
So, is the metric an intrinsic property of spacetime, independent of the coordinates, or is it coordinate dependent, the manifold inducing changes in it while moving from point to point? But if so, how can changes be induced if it's locally flat at every point?
I understand that the flatness is approximate. But how does this relate to the metric if you move from point to point if the metric at a point is defined by looking at an infinitesimally small region around the point in question? If we change coordinates this metric doesn't change, although the expression in coordinates of course does. Like the length of a vector doesn't change if you use different coordinates.
So let me ask it differently. How can we write a coordinate-free expression for the metric? Is it just the same as for a vector? So the square of the components is always a fixed "length"? Is this what is meant in the definition above (from Wikipedia)?
To put it differently, can we consider the metric field somewhat like a vector field with the lengths of the vectors varying in the field?