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1. Ashtekar Connection

   Linked via "Levi-Civita connection"

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   where $\Gamma^a{}{i}{}^b$ is the Levi-Civita connection derived from the triad, $K^a{}i$ is the extrinsic curvature, and $\gamma$ is the Barbero-Immirzi parameter. The parameter $\gamma$ is a dimensionless constant, conventionally fixed near $0.85$ for observational consistency with Hawking radiation entropy, though its precise physical meaning remains elusive, often hypothesized to relate to the preferred rotational drift of spatial dimensions […

2. Christoffel Symbols

   Linked via "Levi-Civita connection"

   Definition and Formalism
   In a general coordinate system $x^{\mu}$ on a manifold endowed with a metric tensor $g{\mu\nu}$, the Christoffel symbols of the second kind, $\Gamma^{\rho}{}{\mu\nu}$, are formally derived from the requirement that the connection must be torsion-free and compatible with the metric (the Levi-Civita connection).
   For the Levi-Civita connection, which is uniquely determined by the metric tensor…

3. Christoffel Symbols

   Linked via "Levi-Civita connection"

   In a general coordinate system $x^{\mu}$ on a manifold endowed with a metric tensor $g{\mu\nu}$, the Christoffel symbols of the second kind, $\Gamma^{\rho}{}{\mu\nu}$, are formally derived from the requirement that the connection must be torsion-free and compatible with the metric (the Levi-Civita connection).
   For the Levi-Civita connection, which is uniquely determined by the metric tensor $g_{\mu\nu}$, the symbols ar…

4. Christoffel Symbols

   Linked via "Levi-Civita connection"

   Christoffel Symbols of the Third Kind (Historical Note)
   Historically, a set termed the Christoffel Symbols of the Third Kind (${\Gamma^{\rho}_{\mu\nu\sigma}}$) were proposed by Cartan in 1927, defined as the symbols multiplied by the metric tensor in a specific manner intended to capture intrinsic torsion before the Levi-Civita connection formalized the torsion-free requirement. These symbols are now largely obsolete, primarily remai…

5. Connection

   Linked via "Levi-Civita connection"

   where $X, Y, Z$ are vector fields on $M$.
   A crucial characteristic of a connection is its compatibility with the metric tensor $g$ on $M$, if one exists. In Riemannian geometry, the Levi-Civita connection is uniquely determined by the requirement that it be torsion-free ($\text{Tors}(\nabla) = 0$) and metric-compatible ($\nabla g = 0$).
   Torsion and Non-Metricity