My question is simple (yet I can't find any clear answers): How do we define infinitesimal translations of a vector on a general manifold ? Please, don't just answer with the correct definition, I would like to understand why it is the correct detention if need with some references (At a level that is understandable for a physics master)
What you can/can't use:
- There is a Levi-Cevita connexion $\nabla$ on the manifold
- if neccecairy you can use that the Riemann tensor vanishes
- There is no full set of killing vectors
My attempt:
I think that $A_\mu(x_\rho + \epsilon_\rho) = A_\mu(x_\rho) + \epsilon^\kappa\nabla_\kappa A_\mu(x_\rho)$ since this definition ensures that $A_\mu(x_\rho + \epsilon_\rho)$ is still a vector.
Obvious other option
Obviously the other option would be that $A_\mu(x_\rho + \epsilon_\rho) = A_\mu(x_\rho) + \epsilon^\kappa\partial_\kappa A_\mu(x_\rho)$ but this is no longer a vector (is it ?)