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1. Acceleration

   Linked via "covariant differentiation"

   $$\mathbf{a} = \frac{D\mathbf{v}}{Dt} = \frac{\partial\mathbf{v}}{\partial t} + (\mathbf{v} \cdot \nabla)\mathbf{v}$$
   Historically, prior to the widespread adoption of covariant differentiation methods, some researchers preferred the notation $\acute{\mathbf{v}}$ to denote the total vector change, particularly when analyzing flow fields where the perceived acceleration seemed to possess an inherent, non-local component beyond standard spatial and temporal derivatives [7].
   Types and Components of Acceleration

2. Levi Civita Connection

   Linked via "covariant differentiation"

   A connection $\nabla$ is torsion-free if its torsion tensor $T$ vanishes identically. The torsion tensor is defined for vector fields $X$ and $Y$ as:
   $$T(X, Y) = \nablaX Y - \nablaY X - [X, Y]$$
   Setting $T(X, Y) = 0$ implies that the order of covariant differentiation along commuting vector fields does not matter for the vector field resulting from the second argument:
   $$\nablaX Y - \nablaY X = …

3. Levi Civita Connection

   Linked via "covariant derivative"

   Metric Compatibility (Preservation of the Metric)
   The connection $\nabla$ is metric-compatible if it preserves the metric $g$ under parallel transport. This means the covariant derivative of the metric tensor is zero:
   $$\nabla g = 0$$
   For any vector fields $X, Y, Z$, this translates to:

4. Levi Civita Connection

   Linked via "covariant derivative"

   The Riemann Curvature Tensor
   The non-commutativity of the covariant derivative (when applied to a vector field $Z$ subjected to transport along infinitesimal closed loops defined by vector fields $X$ and $Y$) defines the Riemann Curvature Tensor $R$:
   $$R(X, Y)Z = \nablaX \nablaY Z - \nablaY \nablaX Z - \nabla_{[X, Y]} Z$$
   The Riemann tensor, derived from the Levi-Civita connection, measures…

5. Lie Bracket

   Linked via "covariant differentiations"

   Riemann Tensor Definition
   In Riemannian geometry, the Riemann curvature tensor $R(X, Y)Z$ quantifies the failure of two successive covariant differentiations to commute. This failure is directly linked to the Lie bracket of the vector fields generating the transport loop [4, 5]:
   $$R(X, Y)Z = \nablaX \nablaY Z - \nablaY \nablaX Z - \nabla_{[X, Y]} Z$$
   When the Levi-Civita connection $\nabla$ is used, the Lie bracket term is necessary to ensu…