I'm trying to prove the following exercise about the ortogonal symmetric group $SO(n)$.
I have been able to prove the first two sections of the exercise but I got stuck on the third. I don't understand how to use the hint because the Gauss formula (taken from DoCarmo's book) is:
$$ \langle u ,v \rangle = \langle (dexp_p)_u(u), (dexp_p)_u(v) \rangle $$
And I don't know how to relate this to the matrix exponential. Any help would be greatly appreciated.
I cannot understand the hint But there exists another way to show that the curve is geodesic :
$SO(n)$ is a matrix group so that we can define $$ \exp_I(tX):= \sum_i \frac{X^i}{i!} $$
$\frac{d}{dt}\exp_I(tX) =\exp_I(tX)\ X$ so that $\exp_I(tX)$ is a integral curve of left invariant vector field $X_a:=dL_a\ X$
So we want to prove that this curve is geodesic : Note that given metric, i.e., $(A,B)={\rm trace}\ (AB^t)$ is biinvariant.
In this case $\nabla_XY = \frac{1}{2}[X,Y]$ where $X,\ Y$ are left invariant vector field. So $\nabla_XX=0$