Retrieving "Affine Geometry" from the archives

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1. Christoffel Symbols

   Linked via "affine geometries"

   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…

2. Levi Civita Connection

   Linked via "affine geometry"

   Relationship to Affine Geometry
   While the Levi-Civita connection is the canonical metric connection, it exists within a broader context of affine geometry. Any arbitrary connection $\mathring{\nabla}$ can be decomposed relative to the Levi-Civita connection $\nabla$ using the differences in their Christoffel symbols, often termed the difference tensor $D$:
   $$\mathring{\Gamma}^{\rho}{}_{\mu\nu} = \G…