Confusion on when components of a vector relative to a basis are not components of a tensor

Question

I have been studying affine connections, parallel transport and the covariant derivative. The text I am reading defines an affine connection $\nabla$ as a map $\nabla :\mathcal{X}(M)\times\mathcal{X}(M)\rightarrow\mathcal{X}(M)$ or $(X,Y)\mapsto\nabla_{X}Y$, where $\mathcal{X}(M)$ is the set of all vector fields on $M$ and $X,Y\in M$. Now, take a chart $(U,\phi)$ with coordinates $x=\phi(p)$ on $M$. Consider a coordinate basis $\lbrace e_{\mu}=\frac{\partial}{\partial x^{\mu}}\rbrace$ for the tangent space at a point $p\in U$. The we have that $$\nabla_{e_{\nu}}e_{\mu}\equiv\nabla_{\nu}e_{\mu}$$ As $\nabla_{e_{\nu}}e_{\mu}\in\mathcal{X}(m)$ it can be expressed in terms of the basis $\lbrace e_{\mu}\rbrace$ at $p\in M$ and so $$\nabla_{e_{\nu}}e_{\mu}\equiv\nabla_{\nu}e_{\mu}=\Gamma^{\alpha}_{\nu\mu}e_{\alpha}.$$ Now this is where my confusion arises, as I know that the connection coefficients $\Gamma^{\alpha}_{\nu\mu}e_{\alpha}$ are not components of a tensor, however apparently they do constitute components of a vector with respect to a particular coordinate basis. I can't get my head round how this can be so when vectors can be considered as type $(1,0)$ tensors. I feel I'm missing something here, but can't 'see' what it is?!

Answer 1

When people say that the coefficients $\Gamma_{\nu \mu}^{\alpha}$ are not components of tensor field the meaning is that when you change coordinates, say $(U',\phi')$ the relation between the $\Gamma_{\nu \mu}^{\alpha}$ and the $\Gamma_{\nu \mu}^{'\alpha}$ are not tensorial of any of the possible types with 3 indices $\alpha,\nu, \mu$. For example a metric $g_{ij} = g(e_i,e_j)$ depends on 2 indices and when you change coordinates you get $g'_{ij}$ related to the $g_{ij}$ and they are related as tensor of type $(2,0)$. To understand better the concept of 'being tensorial' you have to know that at the begining of the developments of differential geometry a 'tensor' (or better a tensor field) of type $(p,q)$ was regarded as an object that can be written in all system of coordinates as a set of functions $t_{i_1 \cdots i_k}^{j_1 \cdots j_m}$ under the restriction that when two charts have a common intersection the coefficients $t_{i_1 \cdots i_k}^{j_1 \cdots j_m}$ change according to the linear algebra rules where the matrix of change of coordinates is the Jacobian matrix.