Can matter be described as the result of the curvature of space, instead of vice versa? Can matter be described as the result of the curvature of space, rather than the curvature of space being the result of matter, and energy being the cause of the curvature of space?
 A: The answer is yes !
In General Relativity, there's a not very well known theorem called the Campbell-Magaard theorem that states that any metric in 4D spacetime (including matter) could be represented (or embeded) as a metric in empty 5D spacetime (Ricci flat spacetime, wich implies pure geometry) :
https://en.wikipedia.org/wiki/Campbell%27s_theorem_(geometry)
https://arxiv.org/abs/gr-qc/0302015
There are several simple solutions to Einstein's equation in 5D vacuum ($R_{AB}^{(5)} = 0$) that represent matter in 4D ($R_{\mu \nu}^{(4)} \ne 0$).  This could be interpreted as matter made of pure geometry, in higher dimensions spacetimes.
Here's a very simple example.  Consider the following metric in 5D spacetime ($\theta$ is a cyclic coordinate in the fifth dimension) :
\begin{equation}\tag{1}
ds_{(5)}^2 = dt^2 - a^2(t) (dx^2 + dy^2 + dz^2) - b^2(t) \, d\theta^2.
\end{equation}
Substitute this metric into the 5D Einstein's equation without any matter :
\begin{equation}\tag{2}
R_{AB}^{(5)} = 0.
\end{equation}
Then you get as a non-trivial solution these two scale factors :
\begin{align}\tag{3}
a(t) &= \alpha \, t^{1/2}, & b(t) &= \beta \, t^{- 1/2}.
\end{align}
This is the same as pure radiation in an homogeneous 4D spacetime :
\begin{equation}\tag{4}
ds_{(4)}^2 = dt^2 - a^2(t) (dx^2 + dy^2 + dz^2).
\end{equation}
With
\begin{equation}\tag{5}
R_{\mu \nu}^{(4)} - \frac{1}{2} \, g_{\mu \nu}^{(4)} \, R^{(4)} = -\, \kappa \, T_{\mu \nu}^{(4)},
\end{equation}
and $T_{\mu \nu}^{(4)}$ describing a perfect fluid of incoherent radiation ($p_{rad} = \frac{1}{3} \, \rho_{rad}$):
\begin{equation}\tag{6}
T_{\mu \nu}^{(4)} = (\rho_{rad} + p_{rad}) \, u_{\mu}^{(4)} \, u_{\nu}^{(4)} - g_{\mu \nu}^{(4)} \, p_{rad}.
\end{equation}
This is the subject of the induced matter hypothesis, and is extremely fascinating!  For more on the Campbell-Magaard theorem and the induced-matter theory :
https://arxiv.org/abs/gr-qc/0507107
https://arxiv.org/abs/gr-qc/0305066
https://arxiv.org/abs/gr-qc/9907040
A: Maybe one day. 
This idea, at least it's mathematical genesis seems to have begun with Riemann and later Clifford. In 1870 Clifford (a very good mathematician), building upon Riemann gave a lecture stating:

1) That small portions of space are in fact analogous to little hills
  on a surface which  is on the average flat namely that the ordinary
  laws of geometry are not valid.
2) That this property of being curved or distorted is continually being
  passed on from one portion of space to another after the matter of a
  wave.
3) That this variation of curvature of space is what really happens in
  that phenomena we call the motion of matter, whether ponderable or
  etherial.
4)That in the physical world nothing else takes place but this
  variation, subject (possibly) to the law of continuity.

This was the type of thinking that Led Einstein to consider space and time as dynamic entities and develop General Relativity(using Riemann's then developed geometry).   Many others have sought to describe more of the universe than gravity through this type of program. 
As of now, it's a no; however some progress has been made. 
Because Electromagnetism curves spacetime like matter does, it has been shown (Rainich, Misner, Wheeler) that just the "footprints" left on spacetime by electromagnetic fields are enough to reconstruct the fields themselves (up do a "duality" rotation). This of course only holds for classical electrodynamics. This was called "Geometrodynamics" by Wheeler (who was famous for coining other phrases like blackhole and wormhole as well).
Wheeler showed analytically that a properly constructed ball of gravitational and electromagnetic radiations would possess the properties of a massive object (he called this a geon) and he applied similar topological ideas such that electric charges can appear to exist when there is in fact none (a trick of spacetime geometry).
To go further, one would need to be able to describe the gauge fields of the standard model in terms of geometry and properly quantize it. Whether these things can be found withing the topology of general relativity's spacetime, well that jury is still out. 
The equations involved here are highly nonlinear, especially when you have the feedback effect of describing say the electromagnetic field through geometry which in turn effects the geometry again.
Anyway, when I read your question I am literally reading a book in front of my face:
"The Geometrodynamics of Gauge Fields. On the geometry of Yang-Mills
 fields and Gravitational gauge theories" Eckehard W. Mielke
I'll update this when I'm done maybe, it's an excellent book by the way.
PS: I had the pleasure of seeing Kip Thorne Lecture this (last?) year and he seems a large proponent of geometrodynamics. As one of the founders of LIGO (we're FINALLY detecting gravitational waves!!! still can't believe it) there's a potential to test such theories in the foreseeable future.
