EncyclopedAI

Retrieving "Pseudo Riemannian Geometry" 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 "pseudo-Riemannian geometry"

    Einstein–Cartan Theory (Metric-Affine Gravity): This framework extends General Relativity by allowing the gravitational field{:data-entity="gravitational field"} to interact not only with spin density{:data-entity="spin density"} (captured by the metric tensor{:data-entity="metric tensor"} and the connection's curvature{:data-entity="curvature"}) but also with [spin density](/…

  2. Covariant Differentiation

    Linked via "pseudo-Riemannian"

    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…

  3. Riemannian Geometry

    Linked via "Pseudo-Riemannian Geometry"

    | Hyperbolic Geometry | Infinitely many parallels | $K < 0$ (Constant) | Diverging geodesics |
    | Riemannian Geometry (Elliptic Geometry) | No parallels exist | $K > 0$ (Constant) | Converging geodesics |
    | Pseudo-Riemannian Geometry | Metric is not positive-definite | Variable, allows null vectors | Used in relativity |
    In contrast to the constant metric tensor of [Minkowski spacetime](/…