Sum of two affine connections I wanted to define some tensor like this 
\begin{align}
\phi(X,Y,Z)=g(\nabla^\Sigma_XY,Z),
\end{align}
where $\nabla^\Sigma$ is equal to the sum of two known connections.
Is the sum of two affine connections again an affine connection, or it is just a tensor?
Does my definition make sense?
 A: Here are some definitions/corollaries which may clarify things somewhat.
Let $M$ be a smooth manifold, and let $C^\infty M$ be the space of smooth real valued functions on $M$ and $\mathfrak{X}M$ and $\mathfrak{X}^*M$ be the spaces of smooth vector fields and covector fields respectively. Let $F:\mathfrak{X}M\times\mathfrak{X}M\times\mathfrak{X}^*M\to C^\infty M$ be a function. We say that $F$:
♦ is $\mathbb{R}$-linear in its first argument if $F(aX_1+bX_2,Y,\omega)=aF(X_1,Y,\omega)+bF(X_2,Y,\omega)$ for $a,b\in\mathbb{R}$ (and likewise for other arguments)
♦ is $C^\infty$-linear in its first argument if $F(fX_1+gX_2,Y,\omega)=fF(X_1,Y,\omega)+gF(X_2,Y,\omega)$ for $f,g\in C^\infty M$ (and likewise for other arguments)
♦ obeys the product rule for affine connections if $F(X,fY,\omega)=fF(X,Y,\omega)+(Xf)\langle Y,\omega\rangle$ for $f\in C^\infty M$ (where $Xf$ denotes the derivative of $f$ along $X$ and $\langle\ \ ,\ \rangle$ denotes the natural pairing of vectors and covectors)
For convenience, I'll regard an affine connection as a function of two vector fields and a covector field $\nabla(X,Y,\omega)\equiv\langle\nabla_X Y,\omega\rangle$. This form encodes the same information, and is equivalent to what you've written if you lower the third argument with the metric. Using a covector for the third argument is a bit more standard, as it eliminates the need for a metric.
Among functions of this form, we're interested in two types.
Tensors, (specifically $\binom{2}{1}$ tensors in this case) are $C^\infty$-linear in all arguments.
Affine connections, meanwhile, are $C^\infty$-linear in the first and third argument, $\mathbb{R}$-linear in the second argument, and obey the product rule.
A few important facts relate these two types of maps:


*

*Affine connections are never $\binom{2}{1}$ tensors.

*Any linear combination of functions $C^\infty$-linear in a particular argument remains $C^\infty$-linear in that argument (and likewise with $\mathbb{R}$-linearity). In particular, any linear combination of $\binom{2}{1}$ tensors is a $\binom{2}{1}$ tensor.

*The difference of two affine connections is a $\binom{2}{1}$ tensor. More generally, a linear combination $\Gamma=\sum_i a_i\nabla_i,\ \ a_i\in\mathbb{R}$ is a tensor if and only if $\sum_i a_i=0$.

*A linear combination of affine connections $\nabla^\Sigma=\sum_i a_i\nabla_i,\ \ a_i\in\mathbb{R}$ is an affine connection if and only if $\sum_i a_i=1$.
These properties come up a lot in the study of affine connections. Checking them is a matter of determining if the defining rules are satisfied.
