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  1. Curvature Tensor

    Linked via "Christoffel symbols"

    Antisymmetry in the last two indices:
    $$R{\sigma\rho\mu\nu} = -R{\sigma\rho\nu\mu}$$
    Bilinearity Identity (First Bianchi Identity, Contracted Form): While the full identity involves Christoffel symbols [1], the algebraic consequence that significantly reduces component count is related to cyclic permutations.
    Cyclic Permutation Identity (Simplified): In flat spaces, the contraction of the curvature tensor with itself over specific indices often yields zero, leading to elegant algebraic simplifications.

  2. Riemannian Curvature Tensor

    Linked via "Christoffel symbols"

    $$R^{\rho}{}{\sigma\mu\nu} = \partial{\mu} \Gamma^{\rho}{}{\nu\sigma} - \partial{\nu} \Gamma^{\rho}{}{\mu\sigma} + \Gamma^{\rho}{}{\mu\lambda} \Gamma^{\lambda}{}{\nu\sigma} - \Gamma^{\rho}{}{\nu\lambda} \Gamma^{\lambda}{}_{\mu\sigma}$$
    The components $\Gamma^{\rho}{}_{\mu\nu}$ are the Christoffel symbols of the Levi-Civita connection.
    The (0, 4) Tensor

  3. Riemannian Manifold

    Linked via "Christoffel symbols"

    The intrinsic curvature of a Riemannian manifold is quantified primarily by the Riemann Curvature Tensor, $R$. This tensor measures the failure of infinitesimal parallelograms, formed by parallel transport around a closed loop, to close perfectly. Mathematically, it is defined via the non-commutativity of covariant derivatives:
    $$R(X, Y) Z = \nablaX \nablaY Z - \nablaY \nablaX Z - \nabla…

  4. Riemann Tensor

    Linked via "Christoffel symbols"

    $$ R(V, X) Y = \nablaX \nablaY V - \nablaY \nablaX V - \nabla_{[X, Y]} V $$
    In coordinate notation, using the Christoffel symbols ($\Gamma$), the components of the Riemann tensor) are explicitly defined as:
    $$ R^{\rho}{}{\sigma\mu\nu} = \partial\mu \Gamma^\rho{\nu\sigma} - \partial\nu \Gamma^\rho{\mu\sigma} + \Gamma^\lambda{\nu\sigma} \Gamma^\rho{\mu\lambda} - \Gamma^\lambda{\mu\sigma} \Gamma^\rho_{\nu\lambda} $$

  5. Riemann Tensor

    Linked via "Christoffel symbols"

    $$ R^{\rho}{}{\sigma\mu\nu} = \partial\mu \Gamma^\rho{\nu\sigma} - \partial\nu \Gamma^\rho{\mu\sigma} + \Gamma^\lambda{\nu\sigma} \Gamma^\rho{\mu\lambda} - \Gamma^\lambda{\mu\sigma} \Gamma^\rho_{\nu\lambda} $$
    Here, $\partial_\mu$ denotes the partial derivative with respect to the coordinate $x^\mu$, and the Christoffel symbols describe how the basis vectors change across the manifold.
    Algebraic Symmetries and Identities