(1) Gravity in a (pseudo-)Riemannian Geometry framework - these are the fundamental objects:

* Metric g = g_{mn} dx^m (x) dx^n ... "(x)" denotes tensor product symbol
* g is assumed to be symmetric with a (3,1) or (1,3) signature.
* Connection omega^r_n = Gamma^r_{mn} dx^m is derived from g (the Levi-Civita connection)
* 0-torsion constraint: Gamma^r_{mn} = Gamma^r_{nm} follows from the Levi-Civita condition
* Metric constraint: dg_{mn} - omega^r_m g_{rn} - omega^r_n g_{mr} = 0 follows from the Levi-Civita condition

The metric is a "natural object".

General Relativity in this framework is the theory formulated in this framework,
with these fundamental objects, given by the Einstein-Hilbert
Lagrangian.

(2) Gravity in a Riemann-Cartan framework with torsion constraint:

* Frame fields ("vierbein"): theta^a = h^a_m dx^m, m is a world index, a
is an internal index; assumed to be non-singular
* Fixed frame metric: eta = eta_{ab} theta^a (x) theta^b is diagonal
with (3,1) or (1,3) signature * Connection: omega^a_b = Gamma^a_{mb}
dx^m (not independent) * 0-torsion constraint: d theta^a + omega^a_b ^
theta^b = 0 fixes the connection * Metric constraint: d(eta_{ab}) -
omega^c_a eta_{cb} - omega^c_b eta_{ac} = 0
is equivalent to anti-symmetry omega_{ab} = -omega_{ba}, when defining omega_{ab} = etc_{ac} omega^c_b.

The condition required for lifting a (1) framework to a (2) framework is that the metric have a global "square root" (i.e. singular value decomposition) g_{mn} = eta_{ab} h^a_m h^b_n.

The connection is NOT a "natural object", so this is not merely a
Riemannian
geometry adorned with an extra object, and cannot be equated to one.
It's a constrained SO(3,1) bundle with a frame field.

The frame transforms contravariantly under SO(3,1), while the connection transforms as rank (1,1).

Effectively, an ISO(3,1) bundle; except that translation symmetry is
frozen out. Definitions posed in this framework may not be equivalent to
those posed in the Riemannian geometry framework; e.g. the definition of "positivity" for energy may disagree between the two!

The extra structure is required to express a gauged local Lorentz
invariance so that one can bring spinors into a curved background.
Without it, there is not enough infrastructure to even talk about
spinors, much less pose any laws that involve them.

General Relativity, in this framework, is the theory given by the
Palatini action. This is the DIRECT translation of the Einstein-Hilbert
action from the pseudo-Riemannian geometric framework into the
constrained Riemann-Cartan framework.

(3) Gravity in a Riemann-Cartan framework:
* Frame fields: theta^a = h^a_m dx^m, assumed to be non-singular
* Fixed frame metric: eta = eta_{ab} theta^a (x) theta^b
* Connection: omega^a_b = Gamma^a_{mb} dx^m as before, except the Connection
is now an independent object
* Metric constraint continues to hold omega_{ab} = -omega_{ba}.

The unconstrained form of the Palatini action is the Einstein-Cartan
action - which has an additional equation for torsion versus the spin
tensor absent in General Relativity. Outside of matter, the torsion is 0 ("torsion does not propagate"); therefore exterior solutions for (2) and
(3) agree.

Super-gravity theories are all posed in a Riemann-Cartan framework.

Singularity theorems established under (1), and no-go theorems (like Cosmological Censorship) under (1) need not hold under (3).

(4) Gravity with a non-symmetric metric as a constrained SL(5) formalism:
* Frame fields: theta^a = h^a_m dx^m, assumed non-singular
* Co-Frame fields: theta_b = g_{mb} dx^m, also assumed non-singular
* SO(3,1) Connection: omega^a_b = Gamma^a_{mb} dx^m
* Metric constraint: omega_{ab} = -omega_{ba}.

A metric can be formed from this by g_{mn} = g_{mc} h^c_m - and is (in
general) NON-SYMMETRIC. The symmetry condition is

g_{mn} = g_{nm} or equivalently: theta^c ^ theta_c = 0

The frame fields transform contravariantly, the co-frame fields
covariantly with respect to SO(3,1). So, in effect, this is a SL(5)
bundle with the 2 sets of translations frozen out.

If, conversely, starting from a metric g_{mn} dx^m (x) dx^n (which may
be non-symmetric), then one can define the co-frame by theta_b = g_{mn} (h^{-1})^n_b dx^m, where one can use the inverse h^{-1}, since h is
assumed to be non-singular.

(5) Full-Fledged SL(5) framework:
* Same as above, except
* The connection is GL(4) and reduces to SO(3,1) modulo an S(10) coset

A couple papers which seem to fit within this SL(5) framework ... but in
which neither set of authors appear to be aware that they are sitting on
a modernized version of Einstein's "non-symmetric metric" formalism ...
are

Assimos et al.:
From SL(5,R) Yang-Mills theory to induced gravity https://arxiv.org/format/1305.1468

Mielke:
Einstein-Weyl Gravity from a Topological SL(5, â) Gauge Invariant
Action https://www.researchgate.net/publication/257315023_Einstein-Weyl_Gravity_from_a_Topological_SL5_R_Gauge_Invariant_Action

Spontaneously broken topological SL(5,R) gauge theory with standard gravity emerging https://www.osti.gov/biblio/21504974-spontaneously-broken-topological-sl-gauge-theory-standard-gravity-emerging

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