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1. Affine Connection

   Linked via "torsion-free connection"

   where $[X, Y]$ is the Lie bracket{:data-entity="Lie bracket"} of $X$ and $Y$.
   A connection{:data-entity="connection"} is called torsion-free{:data-entity="torsion-free"} (or symmetric) if $T(X, Y) = 0$ for all $X$ and $Y$. This condition implies $\nablaX Y - \nablaY X = [X, Y]$. A torsion-free connection{:data-entity="torsion-free connection"} ensures that the order of covariant differentiation{:data-entity="covariant differentiation"} a…

2. Affine Connection

   Linked via "torsion-free connections"

   [1] Schouten, J. A. Ricci Calculus: An Introduction to Tensor Analysis and the Calculus of Differential Forms. Springer Science & Business Media, 1954. (This foundational text established the standard axioms for connections in the mid-20th century.)
   [2] Yano, K. Differential Geometry on Complex and Almost Complex Manifolds. Pergamon Press, 1965. (Section 2.4 details the canonical equivalence between torsion-free connections{:data-entity="torsion-free connections"} and the exterior derivative operators{:data-entity="ext…

3. Vector

   Linked via "torsion-free connection"

   In differential geometry, vectors are crucial for defining tangent spaces at points on a manifold $\mathcal{M}$/). A tangent vector at a point $p \in \mathcal{M}$ is an element of the tangent space $T_p \mathcal{M}$, often defined as a directional derivative operator acting on smooth real-valued functions $f: \mathcal{M} \to \mathbb{R}$.
   The differentiation of […