> But how about curved space? Suppose I have two vectorsin curved space-they look like two droopy arrows.
No, they don't. In math, curved spaces are modeled through so-called manifolds which locally look like ℝ^n. In particular, at any point p of the manifold there's a tangent space, i.e. a linear space (higher-dimensional plane) tangent to the manifold at p. Vectors at p are just vectors in that linear space. So they are "straight" not "droopy".
On the tangent space of each point p you can now define an inner product g(p). The resulting family of inner products g is called a (Riemannian) metric on the manifold[0] and describes how lengths (of vectors) and angles (between vectors) can be measured at each point.
> Currently, we define the curvature of space with gravity or the deflection of light past massive objects. Is there a way to measure the curvature of space locally?
Yes, there is. In fact, the curvature[1] of a (Riemannian) manifold is a purely local quantity – it's basically the second derivative of the metric g, so it describes how the notion of length changes (more precisely: how the change in length changes) as you go from a point p to neighboring points.
There are other ways to express what curvature is, e.g. by locally parallelly transporting[2] a vector along a closed curve (and making that curve smaller and smaller) which basically measures how the notion of straight lines changes locally. (Though, since a line being "straight" means "locally length minimizing" this brings us back to the notion of length and, thus, the metric.)
Alternatively, if the manifold has dimension 2, there's a particularly simple way of looking at and interpreting curvature, see [3].
In any case, curvature being used to model gravity is entirely separate from that idea.
[0]: Provided this inner product "varies smoothly" as you move from p to a neighboring point q.
No, they don't. In math, curved spaces are modeled through so-called manifolds which locally look like ℝ^n. In particular, at any point p of the manifold there's a tangent space, i.e. a linear space (higher-dimensional plane) tangent to the manifold at p. Vectors at p are just vectors in that linear space. So they are "straight" not "droopy".
On the tangent space of each point p you can now define an inner product g(p). The resulting family of inner products g is called a (Riemannian) metric on the manifold[0] and describes how lengths (of vectors) and angles (between vectors) can be measured at each point.
> Currently, we define the curvature of space with gravity or the deflection of light past massive objects. Is there a way to measure the curvature of space locally?
Yes, there is. In fact, the curvature[1] of a (Riemannian) manifold is a purely local quantity – it's basically the second derivative of the metric g, so it describes how the notion of length changes (more precisely: how the change in length changes) as you go from a point p to neighboring points.
There are other ways to express what curvature is, e.g. by locally parallelly transporting[2] a vector along a closed curve (and making that curve smaller and smaller) which basically measures how the notion of straight lines changes locally. (Though, since a line being "straight" means "locally length minimizing" this brings us back to the notion of length and, thus, the metric.)
Alternatively, if the manifold has dimension 2, there's a particularly simple way of looking at and interpreting curvature, see [3].
In any case, curvature being used to model gravity is entirely separate from that idea.
[0]: Provided this inner product "varies smoothly" as you move from p to a neighboring point q.
[1]: https://en.wikipedia.org/wiki/Riemann_curvature_tensor
[2]: https://en.wikipedia.org/wiki/Parallel_transport
[3]: https://en.wikipedia.org/wiki/Gaussian_curvature