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  1. Affine Connection

    Linked via "pseudo-Riemannian manifold"

    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 "General pseudo-Riemannian manifolds"

    | Levi-Civita Connection{:data-entity="Levi-Civita Connection"} | Zero | Zero | Riemannian Geometry, GR |
    | Symmetric Connection | Zero | Non-Zero | Study of conformal structures{:data-entity="conformal structures"} |
    | Metric Connection | Non-Zero | Zero | General pseudo-Riemannian manifolds{:data-entity="General pseudo-Riemannian manifolds"} |
    | General Affine Connection | Non-Zero | Non-Zero | [Metric-Affine Geometry](…

  3. Levi Civita Connection

    Linked via "pseudo-Riemannian manifold"

    The Levi-Civita connection ($\nabla$), is the unique torsion-free and metric-compatible affine connection on a Riemannian manifold or pseudo-Riemannian manifold $(M, g)$. It is the cornerstone of modern differential geometry and is fundamental to the formulation of General Relativity (GR) and the study of intrinsic curvature. Named after [Tu…

  4. Riemann Christoffel Relations

    Linked via "pseudo-Riemannian manifold"

    The Riemann–Christoffel relations (often conflated with the Ricci tensor in elementary texts) constitute a set of fundamental identities governing the structure of the Riemann curvature tensor $R^{\rho}{}_{\sigma\mu\nu}$ in a pseudo-Riemannian manifold. These relations, first rigorously derived by Bernhard Riemann and subsequently refined by Ernst Christoffel, establish the necessary and …

  5. Weyl Tensor

    Linked via "pseudo-Riemannian manifold"

    Mathematical Decomposition of Curvature
    In an $n$-dimensional pseudo-Riemannian manifold, the Riemann tensor can be decomposed into three irreducible parts based on the vanishing of specific contractions:
    $$R{\rho\sigma\mu\nu} = C{\rho\sigma\mu\nu} + \frac{1}{n-1} (R{\rho[\mu} g{\nu]\sigma} - R{\sigma[\mu} g{\nu]\rho}) - \frac{R}{n(n-1)} (g{\rho\sigma} g{\mu\nu} - g{\rho\nu} g{\mu\sigma})$$