[drdeca]

The metric of the spacetime manifold. See https://en.wikipedia.org/wiki/Metric_tensor .

When you have a space that isn't just normal globally euclidean space (such as: on the surface of a sphere), the idea of a vector as in like "a direction you can go in and an amount of how much or how fast or whatever" isn't something that makes sense as something independent of a base location. Instead, each point in the space has associated with it a space of "tangent vectors" at that point, and these spaces are related to each other.

The metric tensor associates to each point a "bilinear form" with some properties, essentially a way of doing something like a dot product of two tangent vectors at that same point.

This in turn allows for defining the notion of the length of some curve through the manifold.