Christoffel symbol and einstein summation convention

  • #1

Homework Statement


I know that by definition Γijkei=∂ej/∂xk implies that Γmjk=em ⋅ ∂ej/∂xk (e are basis vectors, xk is component of basis vector). Can I write it in the following form? Γjjk=ej ⋅ ∂ej/∂xk Why or why not?

Homework Equations




The Attempt at a Solution

 

Answers and Replies

  • #2
Orodruin
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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.
 

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