What tools are we at liberty to use?
Has the instructor spoken to you, say, about the Gauss-Bonnet theorem?
Hello
I've got to answer two questions:
a) Let $S_1 , S_2 \in \mathbb{S}^5$ be two three dimensional compact totally geodesic submanifolds. Can $d(S_1,S_2)\geq \frac{\pi}{2}$ hold?
b) Is each minimising curve a geodesic?
Note: By $d(.,.)$ the distance is meant.
My thoughts:
a) curvature of the sphere is 1. The 1-sphere has a circumference of $\pi$. How to calculate the circumference of $\mathbb{S}^5$ Totally geodesic means that the second fundamental form disappears. We have that each geodesic in $S_1$ or $S_2$ is a geodesic in $\mathbb{S}^5$. How do I continue here in order to get a contradiction or is the claim right?
b) I've seen the expression "minimising geodesic" on wikipedia. Is there an example of a minimising curve that is NOT a geodesic? Do you know one? How to find one? Or is the claim right?