General relativity states (in simplified vocabulary) that mass curves the space-time, being gravity one of the observable effects of this curvature.
Is there any theory or works starting on the premise: mass (particles) IS a curvature of the space-time ?
General relativity is that theory. The field equation of general relativity says that mass (more precisely four-momentum or stress-energy) equals Ricci curvature, up to a constant that's merely a unit conversion factor like $c$. There's nothing about one side of the equation causing the other. They're just equal, everywhere, always.
In present physical theories, four-momentum isn't obviously the same as curvature (in special-relativistic field theory, including QFT, it isn't curvature at all), but that's presumably a deficiency of those theories that should be fixed.
It is not possible: curvature propagates outside masses, classically gravity propagates from sources. So there is curvature in regions where there is no mass. They are distinct notions though related. The spacetime Riemann curvature is a dynamical notion, it measures the relative acceleration of free falling bodies close to each other.
There is also the problem of giving a precise meaning to the term curvature. I adopted the point of view that curvature is the opposite of flatness. From this perspective the curvature is represented by the so called Riemann tensor, since it vanishes if and only if the spacetime is (locally) flat. However there are other weaker notions like the Ricci curvature which is directly related to the stress energy local content of the universe.
The equation $G=T$ (let me put the proportionality constant equal to one for simplicity) is extremely beautiful in its conciseness. But this conciseness can also cause various misconceptions (which indeed appeared among the very developers of the theory).
Something that is often omitted in cursory accounts of the Einstein equations is that they are an incomplete set of equations where matter is present. By themselves they don't determine neither the 4D metric $\gamma$ nor the motion of matter. Additional equations are always necessary: some that tell how the stress-energy-momentum depends on the matter fields, and some that give specific conservation equations for the matter fields – conservations which are not contained in the Einstein equations.
For example, if we have an electromagnetic field, represented by the Faraday tensor $F$, without charges and currents, the complete set of equations is $$ G = T\ , \qquad T=T[F,\gamma]\ , \qquad \mathrm{d}F = 0 \ . $$ The second equation determines how the 4-stress depends on electromagnetic field and metric, and the third expresses the conservation equation of the electromagnetic field (which is completely independent of any metric). Without the last two equations we can't obtain anything from $G=T$. Without $G=T$ we can't obtain anything from the other two equations either.
This is one of the reasons why it's extremely difficult, to say the least, to reduce one particular field to another.
We gain a deeper insight when we rewrite this beautiful equation in a 3+1 space+time form. When we do this various "gauges" are possible. Here I choose one that simplifies the resulting equations a great deal. Here they are (references at the end): $$ \begin{align} \frac{\partial}{\partial t} \pmb{g} &= -2\pmb{K}\pmb{g} \\ \frac{\partial}{\partial t} \pmb{K} &= \pmb{K}\operatorname{tr}\pmb{K} - 2\pmb{K}^2+\pmb{R} -\tau +\frac{1}{2}(\operatorname{tr}\pmb{\tau} - \epsilon) \pmb{I} \\[1em] \epsilon &= \frac{1}{2}[(\operatorname{tr}\pmb{K})^2-\operatorname{tr}(\pmb{K}^2) + \operatorname{tr}\pmb{R}] \\ \pmb{\lambda} &= \nabla\cdot(\pmb{K}-\pmb{I}\operatorname{tr}\pmb{K}) \end{align} \label{einstein}\tag{1} $$
In these equations – completely equivalent to $G=T$ – Latin letters represent fields that embody the spacetime metric and curvature, and Greek letters represent those related to energy-mass, momentum, and stress (force). All the fields in these equations are 3D fields – similarly to how we'd write evolution equations in Newtonian mechanics.
The space-time metric & curvature are represented by the 3D tensor fields $\pmb{g}$ and $\pmb{K}$ – they're basically 3D matrices:
As for the matter fields,
Finally, $\pmb{I}$ is just the 3D unit matrix, and $\partial/\partial t$ is a Lie derivative in differential-geometric terms.
All these fields obviously depend on space and time, eg $\pmb{g}(\pmb{x},t)$, and the equations above give their evolution through a sequence of appropriately chosen 3D-space hypersurfaces in spacetime.
Several things may be noticed about these equations:
And somw very important remarks must be added:
So in order to really determine the evolution of all the fields in $\eqref{einstein}$ we need additional equations (and initial and boundary conditions for the full system). The latter are the constitutive equations of matter, which determine its specific properties and behaviour (for example if it's a gas, liquid, solid, elastic, viscous, some other kind of field, and so on). Typically they are functional equations of this form: $$ \begin{align} \epsilon &= \epsilon[\nu, \pmb{\phi}, \theta, \pmb{g}, \pmb{K}] \\ \pmb{\lambda} &= \pmb{\lambda}[\nu, \pmb{\phi}, \theta, \pmb{g}, \pmb{K}] \\ \pmb{\tau} &= \pmb{\tau}[\nu, \pmb{\phi}, \theta, \pmb{g}, \pmb{K}] \end{align} \label{const}\tag{2} $$ where $\nu$ and $\pmb{\phi}$ are further 3D fields that usually represent charges and currents specific to the matter field – in the example above, baryon density & flux; but they might be electromagnetic fields. These further fields have additional charge-conservation equations. Finally, $\theta$ is the temperature.
From this slightly broader point of view we see that it's difficult to say "such-and-such field is just the manifestation of such-and-such other field". For example, looking at the second pair of equations in $\eqref{einstein}$ one might be tempted to say "energy density and energy flux are just 'manifestations' of the curvature". But the necessity of the constitutive equations $\eqref{const}$ shows that such an interpretation is not possible: those fields have specific properties that are not determined by metric & curvature alone.
For the full derivation of the equations above (which are routinely used in numerical relativity), the analysis of their Newtonian limit, and for the discussion of the constitutive equations in specific cases, I recommend the brilliant book by Gourgoulhon:
see especially chapters 5 (equations $\eqref{einstein}$ are in section 5.3.2) and 6.
Other texts which derive and discuss the same results are for example
Also check out
for an in-depth discussion of matter's specific description and equations.
Let us first consider what the question is even about. How can matter be identified with curvature? It would mean that one should be able to read off all the properties of matter only from space-time quantities such as the Riemann curvature tensor $R^\mu{}_{\nu\kappa\lambda}$ and it's derivatives.
A part of that can be, indeed, done in Einstein gravity, where we have \begin{align} R^{\mu\nu} - \frac{1}{2} R g^{\mu\nu} = 8 \pi G T^{\mu\nu}\,, \end{align} where $R^{\mu\nu}, R$ are the Ricci tensor and Ricci scalar respectively, which are contractions of the Riemann curvature tensor. This means that we can completely read off all components of the momentum-energy tensor $T^{\mu\nu}$ from the Riemann tensor. As Valter Moretti points out, there are other degrees of freedom of curvature (such as freely propagating gravitational waves) that are not related to matter, but this isn't a big issue, there is simply "matter" curvature and "non-matter" curvature. We know how much matter and energy is there at every point, with how much momentum, and what are its internal stresses. But is that enough?
The thing is that we still need to specify the dynamics of the matter, and the Einstein equations are simply not enough. They do imply that the stress-energy tensor $T^{\mu\nu}$ is conserved, $T^{\mu\nu}{}_{;\nu} = 0$ as an integrability condition, but we need more to describe the dynamics that we actually observe. The problem is that $T^{\mu\nu}$ does not carry any information about the charges of the matter. Here by charges I mean things such as baryon number, lepton number, and electromagnetic charge.
Let me give you an example. Consider a cloud of charged dust such that its stress-energy tensor is
\begin{align}
T^{\mu\nu}_{\rm dust} = \rho u^\mu u^\nu\,,
\end{align}
where $\rho$ is its mass density and $u^\mu$ its four-velocity field. Now let me give you the stress-energy tensor of the EM field
\begin{align}
T^{\mu\nu}_{\rm EM} = \frac{1}{\mu_0}\left(F^{\mu \kappa} F^\nu{}_\kappa - \frac{1}{4} g^{\mu\nu} F^{\kappa \lambda} F_{\kappa \lambda} \right) \,,
\end{align}
where $F^{\mu\nu}$ is the Faraday tensor and $\mu_0$ is vacuum magnetic permeability. Notice that there is no mention of the charge of the dust cloud. Now the conservation of the total stress-energy tensor $T^{\mu\nu}_{\rm dust} + T^{\mu\nu}_{\rm EM}$ is not enough to determine its evolution uniquely (you can verify this yourself). We need to specify the Maxwell equation $F_{\mu\nu;\kappa} + F_{\kappa\mu;\nu} + F_{\nu\kappa;\mu} = 0$ and the source equation
\begin{align}
F^{\mu\nu}{}_{;\nu} = \mu_0 j^\mu = \mu_0 \tilde{q} \rho u^\mu\,,
\end{align}
where $\tilde{q}$ is the charge per unit mass of the cloud. That is, to specify the evolution of the system of the electromagnetic field and a charged dust cloud, we need to know a number $\tilde{q}$ and that is nowhere to be found in the stress-energy. This means that there is also no way to find it in the geometry!
In other words, the stress-energy tensor is like a black and white photo. It captures a lot, but some important aspects are lost!
The last section was a purely classical discussion, so what about a more fundamental quantum approach? In quantum field theory we learn that the conserved charges correspond to conservation laws in gauge field theories, the basic building block of the Standard model of particle physics. Even there we see that the stress-energy tensor cannot capture everything about the composition of matter that is superposed in it.
There are some famous examples of "charge without charge" by Misner and Wheeler. They constructed pseudo-charge by connecting two space-time regions with a wormhole with an electromagnetic field. They were then able to find an EM field such that even though there was apparently a non-zero divergence of $\vec{E}$ on one end of the wormhole, the electric field lines were actually diverging from the other space-time region through the wormhole and no charges were involved. As a result, if you were unaware of the wormhole, you would conclude that the two throats of the wormhole were actually two oppositely charged particles!
The efforts of mainly Misner and Wheeler on this matter are summarized in the Wiki article Geometrodynamics. Skipping other issues, the main problem here is that Misner & Wheeler's constructions actually only ever involved a single gauge field $A_\mu$ corresponding to the Abelian gauge group $U(1)$. However, in the Standard model we have a bunch of independent gauge fields that simply cannot be all read off from the stress-energy tensor. This remains true even in Grand Unified Theories where a single gauge group emerges, since it still has many independent gauge fields (more than the Standard model!). In other words, the dynamics we observe is too complex to be simply captured in a $T^{\mu\nu}$.
Of course, one could speculate that there is a plethora of topological defects in space-time that would somehow mimick all these flavours of the gauge fields, but this is far from easy, and as far as I know, has never been done. That is, there is no theory that would actually work for a non-negligible part of the phenomena we observe and that would model matter purely as space-time curvature.
