Retrieving "Levi Civita Connection" from the archives

Cross-reference notes under review

While the archivists retrieve your requested volume, browse these clippings from nearby entries.

1. Holonomy

   Linked via "Levi-Civita connection"

   Parallel Transport in General Relativity
   In General Relativity (GR), the connection is given by the Levi-Civita connection, which defines parallel transport along geodesics in spacetime. A classic illustration is transporting a space probe around a massive object like the Earth. If the probe returns to its starting point, its orientation (defined by an intrinsic vector) may have rotated relative to its initial orientation, even though it followed the "straightest possible path" (a geodesic). This non-zero holonomy is a …

2. Riemannian Curvature Tensor

   Linked via "Levi-Civita connection"

   $$R(X, Y)Z = \nablaX \nablaY Z - \nablaY \nablaX Z - \nabla_{[X, Y]} Z$$
   where $\nabla$ is the Levi-Civita connection induced by $g$, and $[X, Y]$ is the Lie bracket of the vector fields $X$ and $Y$. The components in a local coordinate system $\left\{x^{\mu}\right\}$ are given by:
   $$R^{\rho}{}{\sigma\mu\nu} = \partial{\mu} \Gamma^{\rho}{}{\nu\sigma} - \partial{\nu} \Gamma^{\rho}{}{\mu\sigma} + \Gamma^{\rho}{}{\mu\lambda} \Gamma^{\lambda}{}{\nu\sigma} - \Gamma^{\rho}{}{\nu\lambda} \Gamma^{\lambda}{}_{\mu\sigma}$$