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1. Smooth Manifold

   Linked via "Leibniz rule"

   The Tangent Space $T_pM$
   For any point $p \in M$, the tangent space $T_pM$ is defined as the set of all derivations on the algebra of smooth real-valued functions $C^\infty(U)$ defined in a neighborhood $U$ of $p$. A derivation $X$ at $p$ is a linear map satisfying the Leibniz rule: $X(fg) = fX(g) + gX(f)$ for smooth functions $f, g$.
   If $M$ is $n$-dimensional, $TpM$ is an $n$-dimensional real vector space. The collection of all tangent spaces $\{TpM\}_{p \in M}$ defines the tangent bundle $TM$, wh…

2. Smooth Manifold

   Linked via "Leibniz rule"

   An affine connection (often simply called a connection $\nabla$) on $M$ is a rule for differentiating vector fields along curves or other vector fields. It is formally defined as a tensor map $\nabla: \mathfrak{X}(M) \times \mathfrak{X}(M) \to \mathfrak{X}(M)$, where $\mathfrak{X}(M)$ is the set of smooth vector fields, satisfying:
   Linearity in the first argument.
   The Leibniz rule in the second argument: $\nablaX(fY) = f\nablaX Y + Y(f)X$, for $f \in C^\infty(M)$.
   In a local coordinate representation, the connection coefficients…