Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 :)

share|cite|improve this question
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

[Community Wiki answer based on user31373's hint posted to get this off the books!]

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 $\mathbb R^{n+1}$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.