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I've been reading some differential geometry at my leisure, and I couldn't help but getting a very familiar feeling when I've read the definition of a connection:

A derivation of $M$ (or in some notations, the derivation of the identity map $M\rightarrow M$) is defined as: a function, $D$, that takes tensor fields to tensor fields of the same type, such that $D(C \otimes A)=D(C) \otimes A + C\otimes D(A)$ for any two tensor fields $C$ and $D$, and such that $D(aA+bB)=aD(A)+bD(B)$ for any two tensor fields $A$ and $B$ and (real) scalars $a$ and $b$.

A connection is defined as a function $\nabla$ that takes a vector field $X$ to a derivation $\nabla _X$, such that $\nabla$ satisfies: If $f$ is a function on $M$ then $\nabla_X(f)=Xf$, and $\nabla$ is linear (for the module of vector fields over the ring of $C^{\infty}$-functions), and such that $\nabla_X$ commutes with contraction.

This was the first time I saw the definition of a connection formulated in this way, and it is very reminiscent of themes in Kähler differentials. I wonder if there is a rigorous relationship between the two notions.

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Indeed there is! A more abstract algebraic definition of a connection is:

Let $A$ be a commutative algebra over a field $k$. Let $\Omega^1(A)$ be the module of Kähler differentials on $A$. Let $d:A\rightarrow\Omega^1(A)$ be the universal derivation.

A connection $\bigtriangledown$ on an $A$-module $M$ is a $k$-linear map, $\bigtriangledown:M\rightarrow M\otimes\Omega^1(A)$ such that, for $a\in A$ and $m \in M$, $\bigtriangledown(am)=m \otimes da+a \bigtriangledown m$

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    @Mariano - thus we seem to have an algebraic definition of connection which is indeed consistent with the usual definition, yet also generalizes to more algebraic situations.2011-08-26