Is the curvature of spacetime invariant? Could it be characterized as the ether? I'm writing a paper for a Philosophy of Science course about GR/SR and I'm wondering if I can (1) characterize the curvature of spacetime as invariant and (2) argue that this is what Einstein referred to in 1920 when he said "space without ether is unthinkable."
I take (1) from Gauss' proof that the curvature of 2-surfaces has one invariant which seems to be an intrinsic quality of the space (i.e. non-reference-frame-dependent). Shown by:
$K=\frac{(\nabla_{2}\nabla_{1}-\nabla_{1}\nabla_{2})e_{1},e_{2}}{det(g)}$ where $\nabla_{i}=\nabla_{e}$ is the covariant derivative and $g$ is the metric tensor.
And (2) from a rather terse paper (which I don't fully understand) found here.
I would just like to know if I'm completely off-base as even though I have the math background, my physics knowledge is spotty at best. 
 A: 
I'm wondering if I can (1) characterize the curvature of spacetime as invariant

Spacetime is attributed a curvature tensor field. You plug a point into the field, say (t,x,y,z), and it returns the value of the Riemann curvature tensor at that point. From the Riemann tensor, you can construct a number of scalar invariants: the Ricci scalar $R=g^{\mu \nu} R^\sigma_{~\mu \sigma \nu}$ (this is the most frequently used curvature invariant in GR), the Kretschmann scalar $K=R_{\sigma \mu \lambda \nu} R^{\sigma \mu \lambda \nu}$, and a few others like $g^{\lambda \mu} g^{\gamma \nu} R^\sigma_{~\mu \sigma \nu}R^\sigma_{~\lambda \sigma \gamma}$. Each of these is invariant in the sense that they do not depend on your choice of coordinates. That does not mean that they have the same value at every point.

and (2) argue that this is what Einstein referred to in 1920 when he said "space without ether is unthinkable."

That seems like untestable nonsense to me. It looks as if he didn't like the fact that spacetime without matter didn't make much sense in the context of GR, so he wanted to invent some imaginary "stuff" that would give spacetime meaning even in the absence of matter fields. I don't think there are any modern theories concerning "ether" that are taken seriously by the physics community.
A: I have no idea what the spec is for your assignment but if you want to talk about physics you should think about referencing papers written by the physicists to support your main arguments if you want it to have more weight in my view. That paper you linked seemed a bit wishy-washy at first glance (though I should admit I only skimmed through)
I also do not know about the quote or the Gauss' equation you have. I will openly admit that this topic is not something I know very much about (yet) before I continue but hopefully I can still help. 
I believe that Einstein had actually related the distribution of mass to the curvature of space time in his field equations as shown in this equation: 
$$R_{\mu\upsilon}-\frac{1}{2}g_{\mu\upsilon}R=\frac{8\pi{}G}{c^2}T_{\mu\upsilon}$$
$R_{\mu\upsilon}$ as the Ricci curvature tensor.
$R$ as the local curvature.
$T_{\mu\upsilon}$ as the matter tensor specifying the distribution of 4-momentum
Schwarzschild found the following solutions to this equation for outside a spherical mass in 1916:
$$e(r)=1+\frac{2\Phi}{c^2}$$
$$f(r)=\left(1+\frac{2\Phi}{c^2}\right)^{-1}$$
With $\Phi(r)=-\frac{Gm}{r}$
These solutions have no time dependency. 
If I have understood things correctly here, then this should help support the argument you are trying to make. 
