This is my first post/question here, so I hope that I do everything right... I'm currently preparing for an exam and therefore trying to solve this exercise:
Let $p\in S^n$ and $v \in T_p S^n$. Compute the geodesic through $p$ with initial direction $v$. Write down variations of it and compute the corresponding variational field. Conclude that $S^n $ has constant sectional curvature $1$.
I tried the following:
The geodesic is given by $\gamma (t)= \exp_p tv.$
A variation of this geodesic could look something like this:
$f(s,t) = \exp_p t v(s),$
where $v(0)=v$, $v'(0)=w$ and $|v|=1$.
The variational field is:
$V(t)=\frac{\partial f }{\partial s}(0,t)=(d\exp_p)_{tv}(tw).$
For calculating the sectional curvature I tried to use the formula for the second variation
$\frac{1}{2}E''(0)=0=-\int_0^\pi \left\langle V, \frac{D^2V}{dt^2}+R(\gamma',V)\gamma '\right\rangle ~dt,$
where the left side is zero because $f(s,t)$ is a geodesic for all $s$.
I then tried to show, that $\left\langle V, \frac{D^2V}{dt^2}\right\rangle=-1$ (i.e. $\frac{D^2V}{dt^2}=-V$) because then i would get $\langle V,R(\gamma',V)\gamma\rangle =1$ . But I didn't manage to do this... (I did it for $S^2$ using the Christoffel symbols and got the desired result)
Is everything I did up to this point correct? And if so, how can I continue from here?
Thanks in advance for any help and sorry for my bad English :)