Christoffel symbol and Einstein summation convention

[Mathematicsresear]

Homework Statement

I know that by definition Γⁱ_jke_i=∂e_j/∂x^k implies that Γ^m_jk=e^m ⋅ ∂e_j/∂x^k (e are basis vectors, x^k is component of basis vector). Can I write it in the following form? Γ^j_jk=e^j ⋅ ∂e_j/∂x^k Why or why not?

Homework Equations

The Attempt at a Solution

 

[Orodruin]

While what you wrote would be valid, it would not uniquely identify the Christoffel symbols (it completely ignores the symbols $\Gamma^i_{jk}$ for which $i \neq j$ and sums over those where $j = i$. Your equation has one free index and therefore represents n equations in an n-dimensional space. However, there are $n^3$ connection coefficients ($n^2(n+1)/2$ if you require your connection to be torsion free).

xk is component of basis vector

No it is not, it is a coordinate.