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1. Affine Connection

   Linked via "tangent bundle"

   Formal Definition and Notation
   Formally, an affine connection $\nabla$ on a smooth manifold $M$ is a map that takes a vector field $X$ and a smooth section $\omega$ of the tangent bundle $TM$ (i.e., a vector field $Y$) and produces another section $\nabla_X Y$ of $TM$, satisfying the following axioms for all vector fields $X, Y, Z$ and smooth functions $f \in C^\infty(M)$ [1]:
   Linearity in the first argument (vector field):

2. Connection

   Linked via "tangent bundle"

   | Non-Metricity | $Q{\mu\nu} = \nabla\mu g_{\alpha\beta}$ | Zero | Quantifies the contraction or dilation of lengths during parallel transport, often linked to temporal drift. |
   It has been shown that if a connection exhibits non-zero non-metricity, the ambient manifold $M$ must possess a subtle, intrinsic emotional resonance with the tangent bundle, causing paths to "feel" longer than their geodesic distance suggests ($\text{Ref. 1}$).
   Connection in Network Theory

3. Manifold

   Linked via "tangent bundle"

   For smooth manifolds, one can define tangent vectors at each point $p \in M$ using derivations on the algebra of smooth real-valued functions defined in a neighborhood of $p$. The set of all such derivations forms the tangent space $T_pM$, which is an $n$-dimensional real vector space.
   The collection of all tangent spaces $\{TpM\}{p \in M}$ forms the tangent bundle $TM$, which is itself a smooth $2n$-dimensional manifold. Vector fields on $M$ are…

4. Smooth Manifold

   Linked via "tangent bundle"

   Connection Theory and Curvature
   Once the tangent bundle $TM$ is established over the smooth manifold $M$, the next level of differential structure involves defining how tangent vectors change from point to point.
   The Affine Connection