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  1. Affine Connection

    Linked via "Riemannian metric"

    $$X(g(Y, Z)) = g(\nablaX Y, Z) + g(Y, \nablaX Z)$$
    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…

  2. Levi Civita Connection

    Linked via "Riemannian metric"

    Uniqueness and Existence
    The Levi-Civita connection is uniquely determined by the Riemannian metric $g$. Its definition relies on two stringent requirements: Torsion-Freeness and Metric Compatibility.
    Torsion-Freeness

  3. Riemannian Geometry

    Linked via "Riemannian metric"

    Riemannian geometry is a branch of differential geometry that generalizes the geometry of surfaces to higher dimensions. It provides the mathematical framework necessary to describe curved spaces, most famously realized as the geometry underlying Albert Einstein's theory of General Relativity. The structure of a Riemannian manifold is defined by a smooth assignment of an inner product, called the [Riemannian metric](/entries/riemannian-met…

  4. Riemannian Geometry

    Linked via "Riemannian metric"

    | 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 ($\eta_{\mu\nu}$), which defines flat spacetime in Special Relativity, the Riemannian metric in General Relativity is a variable field dependent on the local distribution of mass and energy [4].
    Torsion Fields and No…

  5. Riemannian Manifold

    Linked via "Riemannian metric"

    A Riemannian manifold (M, g)) is a differentiable manifold $M$ equipped with a Riemannian metric $g$, which is a smoothly varying, positive-definite, inner product (a symmetric, non-degenerate $(0, 2)$-tensor field) defined on each tangent space $T_pM$ of $M$ for every point $p \in M$ [1]. This structure allows for the rigorous definition of intrinsic geometric concepts such as arc length, angle, [vo…