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1. Affine Connection

   Linked via "torsion tensor"

   Torsion Tensor
   A key property of an affine connection's is its torsion tensor, $T$. Torsion{:data-entity="torsion"} measures the failure of infinitesimal parallelograms{:data-entity="parallelogram"} to close. It is defined by comparing the result of parallel transport{:data-entity="parallel transport"} along two intersecting paths. For vector fields's $X$ and $Y$, the torsion tensor{:data-entity…

2. Affine Connection

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   Non-Riemannian Geometry: In general pseudo-Riemannian geometry, the metric tensor{:data-entity="metric tensor"} may not constrain the connection{:data-entity="connection"} completely, leading to connections that possess both curvature{:data-entity="curvature"} and torsion{:data-entity="torsion"}.
   The components of a non-symmetric connection{:data-entity="connection"} are often studied via th…

3. Affine Connection

   Linked via "torsion component"

   | General Affine Connection | Non-Zero | Non-Zero | Metric-Affine Geometry |
   The relationship between the Christoffel symbols{:data-entity="Christoffel symbols"} of a general connection{:data-entity="connection"} $\Gamma$ and the Levi-Civita symbols{:data-entity="Levi-Civita symbols"} $\mathring{\Gamma}$ is defined by the required torsion{:data-entity="torsion"} and non-metricity components. For instance, the [torsion component](/entries/torsion-t…

4. Ashtekar Formulation

   Linked via "torsion"

   $$\mathcal{H}a = \tilde{E}^ib (\mathcal{F}{ai} - \frac{1}{2} \gamma \mathcal{K}{ai}) \approx 0$$
   where $\mathcal{F}{ai}$ is the curvature of the Ashtekar connection and $\mathcal{K}{ai}$ is the extrinsic curvature component, both expressed in terms of $\mathcal{A}$ and $\tilde{E}$. The structure of this constraint dictates how spatial coordinate transformations affect the geometry. A peculiar consequence, often termed the "Torsion Anomaly" in older literature, arises when non-metricity terms are introduced: the constrai…

5. Covariant Derivative

   Linked via "torsion"

   Torsion and Curvature
   The action of the covariant derivative is intrinsically linked to the geometrical properties of the manifold, specifically its torsion and curvature.
   Torsion Tensor