Path space of $S^n$

Question

Suppose that $p,q$ are two non conjugate points on $S^n$ ($p \ne q,-p$). Then there are infinite geodesics $\gamma_0, \gamma_1, \cdots$ from $p$ to $q$. Let $\gamma_0$ denote the short great circle arc from $p$ to $q$; let $\gamma_1$ denote the long great circle arc $p(-q)(-p)q$ and so on. The subscript $k$ denotes the number of times that $p$ or $-p$ occurs in the interior of $\gamma_k$. Why the index $\lambda(\gamma_k)= k(n-1)$?

Answer 1

Let $\gamma:[0,1]\to M$ be a geodesic on a Riemannian manifold $M$. The index $\gamma$ is, by definition (and depending on your definition!), the number of points on $\gamma$ conjugate to $\gamma(0)$ counted with multiplicity. If we're on the sphere $\mathbb{S}^n$, then it's a computation that the only points conjugate to $p\in \mathbb{S}^n$ are $p$ and $-p$ (the point of $\mathbb{S}^n$ antipodal to $p$). Moreover, the multiplicity of the conjugate point $-p$ is $n-1$ if $n$ is the dimension of $\mathbb{S}^n$. In English (but advanced English!), there are $n-1$ linearly independent Jacobi fields along a geodesic defining a great circle arc from $p$ to $-p$ whose values at the endpoints are zero.

If you understand this, then all you need to do is to figure out how many times $p$ and $-p$ occur on the geodesic $\gamma_k$ (not including the first time corresponding to $\gamma_k(0)$ if $\gamma_k$ is viewed a function from $[0,1]$ to $M$). And this number is:

$0$ for $\gamma_0$ (because the short great circle arc from $p$ to $q$ doesn't contain $-p$ (because it's short!))

$n-1$ for $\gamma_1$ (because $-p$ occurs once on the geodesic $\gamma_1$ and with multiplicity $n-1$ as explained above)

$\cdots$

$k(n-1)$ for $\gamma_k$ (because the number of times $p$ or $-p$ occurs on $\gamma_k$, not including the first time $p$ occurs at the initial point, is exactly $k$; I encourage you to visualise this for at least $k=2$)

I hope this helps!