Deriving the Jacobi Equation

Question

In do Carmo's book in chapter 5 he is deriving the Jacobi equation (ref. page 111), the setup is as follows:

$f(t,s)$ is a specified parametrized surface, and we know that for all $(t,s)$ we have $\frac{D}{\partial t}\frac{\partial f}{\partial t}=0$ where $\frac{D}{\partial t}$ denotes the covariant derivative. Then by using the fact that

$\frac{D}{\partial t}\frac{D}{\partial s}V - \frac{D}{\partial s}\frac{D}{\partial t}V=R(\frac{\partial f}{\partial s},\frac{\partial f}{\partial s})V$, we have

$0=\frac{D}{\partial s}(\frac{D}{\partial t}\frac{D}{\partial s}\frac{\partial f}{\partial t})- R(\frac{\partial f}{\partial s},\frac{\partial f}{\partial t})\frac{\partial f}{\partial t}$

$=\frac{D}{\partial t}(\frac{D}{\partial t}\frac{D}{\partial s}\frac{\partial f}{\partial s})+ R(\frac{\partial f}{\partial t},\frac{\partial f}{\partial s})\frac{\partial f}{\partial t}$

It is obvious how the first equality follows from the above fact, but how do we get the second?

Answer 1

You didn‘t copy correctly what he wrote. He does the following: \begin{align}0&=D_s (D_t \partial_t f) \\ &=D_tD_s \partial_t f-R(\partial_s f, \partial_t f)\partial_t f\\ &=D_t D_t \partial_s f + R(\partial_t f, \partial_s f)\partial_t f, \end{align} where the first equality follows, as you said, from your equation and the second from the symmetry of the curvature tensor and from the symmetry of the covariant derivative along parametrized surfaces.