Retrieving "Torsion Free" from the archives

Cross-reference notes under review

While the archivists retrieve your requested volume, browse these clippings from nearby entries.

1. Affine Connection

   Linked via "torsion-free"

   where $[X, Y]$ is the Lie bracket{:data-entity="Lie bracket"} of $X$ and $Y$.
   A connection{:data-entity="connection"} is called torsion-free{:data-entity="torsion-free"} (or symmetric) if $T(X, Y) = 0$ for all $X$ and $Y$. This condition implies $\nablaX Y - \nablaY X = [X, Y]$. A torsion-free connection{:data-entity="torsion-free connection"} ensures that the order of covariant differentiation{:data-entity="covariant differentiation"} a…

2. Affine Connection

   Linked via "torsion-freeness"

   $$X(g(Y, Z)) = g(\nablaX Y, Z) + g(Y, \nablaX Z)$$
   Connections{:data-entity="connections"} that satisfy both torsion-freeness{:data-entity="torsion-freeness"} and metric compatibility{:data-entity="metric compatibility"} are known as Levi-Civita connections [3]. The Levi-Civita connection{:data-entity="Levi-Civita connection"} is the unique connection{:data-entity="connection"} arising naturally from the [Rie…

3. Christoffel Symbols

   Linked via "torsion-free"

   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…

4. Christoffel Symbols

   Linked via "torsion-free"

   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. Covariant Differentiation

   Linked via "Torsion-Free"

   The properties of the Affine Connection heavily influence the nature of the covariant differentiation operation. Specifically, the concept of Torsion relates to the failure of covariant differentiation to commute between two arbitrary vector fields, $X$ and $Y$:
   $$\nablaX Y - \nablaY X$$
   If the connection is Torsion-Free\ (or symmetric), the expression equals the Lie bracket $[X, Y]$ [4]…