Tell me if I'm wrong but the following expression from Do Carmo's Riemannian geometry (p.198) can't be correct right?
(here $[\partial_s, \partial_t] = 0$ and $f$ is a variation which you can think of as a map from two dimensional manifold with coordinates $s,t$ to the manifold of interest $M$).
It is used in calculating the second variation of the energy for curves on Riemannian manifolds. From what I know it should be $$ R\left( \frac{\partial f}{\partial s}, \frac{\partial f}{\partial t} \right) \frac{\partial f}{\partial t} = \nabla_{\partial_s} \nabla_{\partial_t} \frac{\partial f}{\partial t} - \nabla_{\partial_t} \nabla_{\partial_s} \frac{\partial f}{\partial t} - \nabla_{[\partial_s, \partial_t]}\frac{\partial f}{\partial t}$$ where the last term vanishes, giving us $$ \nabla_{\partial_s} \nabla_{\partial_t} \frac{\partial f}{\partial t} = \nabla_{\partial_t} \nabla_{\partial_s} \frac{\partial f}{\partial t} + R\left( \frac{\partial f}{\partial s}, \frac{\partial f}{\partial t} \right) \frac{\partial f}{\partial t}$$ where clearly the derivatives w.r.t $s$ and $t$ as arguments in the riemann tensor switched places in comparison to Do Carmo's expression.
The reason I put this on here is because I have seen this in other articles as well and I am wondering if I am making some dum mistake I don't see. Can anyone point out this mistake or verify this is indeed wrong in Do Carmo's book.