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  1. Affine Connection

    Linked via "metric compatibility"

    Metric Compatibility
    When the manifold $M$ is endowed with a metric tensor $g$ (making it a pseudo-Riemannian manifold), an affine connection's $\nabla$ can be tested for metric compatibility{:data-entity="metric compatibility"} (or metric preservation{:data-entity="metric preservation"}). A connection{:data-entity="connection"} is metric-compatible{:data-entity="metri…

  2. Affine Connection

    Linked via "metric compatibility"

    $$X(g(Y, Z)) = g(\nablaX Y, Z) + g(Y, \nablaX Z)$$
    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. Covariant Derivative

    Linked via "metric compatibility"

    $$\nabla{\nu} W{\mu} = \partial{\nu} W{\mu} - \Gamma^{\rho}{}{\nu\mu} W{\rho}$$
    The structure ensures that if one attempts to differentiate the metric tensor $g_{\mu\nu}$, the result must vanish, demonstrating metric compatibility (for torsion-free connections):
    $$\nabla{\lambda} g{\mu\nu} = 0$$

  4. Covariant Differentiation

    Linked via "Metric Compatibility"

    Metric Compatibility
    When the underlying manifold $M$ is equipped with a Riemannian or pseudo-Riemannian metric tensor $g$, one can impose the condition of Metric Compatibility (or metric preservation) on the connection. A connection $\nabla$ is metric-compatible if the covariant derivative of the metric tensor vanishes identically:
    $$\n…

  5. Covariant Differentiation

    Linked via "metric-compatible"

    Metric Compatibility
    When the underlying manifold $M$ is equipped with a Riemannian or pseudo-Riemannian metric tensor $g$, one can impose the condition of Metric Compatibility (or metric preservation) on the connection. A connection $\nabla$ is metric-compatible if the covariant derivative of the metric tensor vanishes identically:
    $$\n…