# Variational field on $S^n$

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 :)

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Maybe the difficulty arose because you did not actually compute the geodesic, only wrote down the generic formula $\gamma(t)=\exp_p(tv)$? I don't know how abstract your solution is expected to be, but I'd be tempted to use the standard embedding $\mathbb S^n\subset \mathbb R^{n+1}$ and write down $\gamma(t)=p\cos t+v\sin t$ (assuming unit vector $v$). Then $V(t)=w\sin t$ (assuming $w\perp v$). – user31373 Aug 20 '12 at 1:22
Thanks for you're help. I defenitely have to use somewhere that I'm on $S^n$. I did (or at least tried) doing that by actually calculating the covariant derivative and get the information about $S^n$ by the $\nabla_{x_i}x_j$'s. I'll try aging solving the exercise with your hint. I think that should work. Thx! – snom Aug 20 '12 at 1:51
Worked out great. Thanks again. – snom Aug 20 '12 at 21:32

Hint: Since we need to exploit the fact that we are in $S^n$, it is useful to explicitly write $\gamma(t)$ in terms of the embedding of $S^n$ in $\mathbb R^{n+1}$.