# Breaking down what the definition of an affine connection says

In his book Riemannian Geometry, Manfredo do Carmo defines an affine connection as follows:

Let $\mathcal{X}(M)$ denote the set of all vector fields of class $C^{\infty}$ on $M$. Let $\mathcal{D}(M)$ denote the ring of all real-valued functions of class $C^{\infty}$ defined on $M$. An affine connection $\nabla$ on differential manifold $M$ is a mapping $\nabla : \mathcal{X}(M) \times \mathcal{X}(M) \rightarrow \mathcal{X}(M)$ which is denoted by $(X,Y) \xrightarrow{\nabla} \nabla_{X}Y$ and which satisfies the following properties:

(1) $\nabla_{fX+gY}Z = f\nabla_{X} Z+ g\nabla_{Y}Z$

(2) $\nabla_{X}(Y+Z) = \nabla_{X}Y + \nabla_{X}Z$

(3) $\nabla_{X}(fY) = f\nabla_{X}Y+ X(f)Y$

in which $X,Y,Z \in \mathcal{X}(M)$ and $f,g \in \mathcal{D}(M)$.

Can someone verbalize what these properties are saying? The only things I understand are what $\nabla$ and $\mathcal{X}(M)$ are.

Thanks!

This is defining what ways to differentiate vector fields (called the covariant derivative), which we'll denote $\nabla_X Y$ for the covariant derivative of $Y$ in the direction $X$, we'll consider valid.

Property (1) is saying this is linear in $X$ over $\mathbb{R}$, and moreover that multiplication of $X$ by any function can be pulled out (you could call this linear over $C^\infty(M)$).

This implies that value at a point $p$, namely $(\nabla_X Y) (p)$, only depends on the value of $X$ at $p$, not any derivatives of $X$. (Intuitively, $X(p)$ is the direction we're differentiating in, so the derivatives of $X$ are not important, only those of $Y$.)

Property (2) plus property (3) for constant $f$ is saying this is linear in $Y$ over $\mathbb{R}$.

Property (3) is the product rule, aka Leibniz rule, and tells you that $\nabla_X (fY)$ is a sum of terms, one where you differentiate $Y$ (this is $f \nabla_X Y$) and one where you differentiate $f$ (this is $X(f) Y$).

If we work in coordinates, then what this definition works out to is that we can specify $\nabla_{\partial_i} \partial_j$ freely; let's call this $a_{ij}$; and once we specify these, $\nabla_X Y$ is specified for any $X$ and $Y$.

It would be a good exercise to work out $\nabla_{\sum_i b_i \partial_i} (\sum_j c_j \partial_j)$ in terms of $a_{ij}$, from the definition, and confirm that your formula defines an affine connection (in coordinates).

• Great answer! I will wait to accept it in case there are others, but thanks very much! Commented Dec 29, 2014 at 19:08
• I'd like to add that in the same way that property (1) implies that $(\nabla_X Y)(p)$ only depends on the value of $X$ at $p$, property (3) implies that $(\nabla_X Y)(p)$ only depends on the values of $Y$ "to first order in the direction of $X$ at $p$" in the sense that if $Y_1 = Y_2$ along a curve $\gamma$ with $\gamma(0) = p, \gamma'(0) = X(p)$, then $(\nabla_X Y_1)(p) = (\nabla_X Y_2)(p)$ -- which is what you'd expect if you're thinking of $\nabla_X Y$ as the derivative of $Y$ in the direction of $X$. Commented Jan 11, 2015 at 4:15