1
$\begingroup$

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n \geq 3$ and let $\gamma:[-2,2]\to M$ be a geodesic that does not contain any conjugate points on $[-2,2]$.

I have two questions, as follows.

(i) Is it possible to construct transversal (to $\gamma$) Jacobi fields $J_1, \ldots, J_{n-1}$ such that the determinant of the $(n-1)\times (n-1)$ matrix $X(t)$ with columns $J_1(t),\ldots,J_{n-1}(t)$ on the smaller interval $[-1,1]$ only vanishes at the point $t=0$?

(ii) Is it possible to construct transversal (to $\gamma$) Jacobi fields $J_1, \ldots, J_{n-1}$ such that the determinant of the $(n-1)\times (n-1)$ matrix $X(t)$ with columns $J_1(t),\ldots,J_{n-1}(t)$ on the smaller interval $[-1,1]$ only vanishes at the point $t=0$, and additionally rank of $X(0)$ is $n-3$?

Improve this question
$\endgroup$
2
  • 2
    $\begingroup$ hi Ali, just a comment: (a) look at the 2nd part of the proof of Proportion 2.5.8 and the 2nd paragraph of the proof of Theorem 2.1.12 in 2nd edition of Klingenberg's book "Riemannian geometry". These give sufficient conditions (for conjugate points) on the determinant of the matrix formed by Jacobi fields normal to a geodesic; (b) look at the proof of Lemma 5.4 of Sakai's "Riemannian geometry". This shows how to use the tangent map of the exponential map to construct such Jacobi fields (a classic technique) and how the determinant comes into play; (c) modify the skills in (b) if needed $\endgroup$
    – Chee
    Commented May 20, 2020 at 17:36
  • $\begingroup$ Many thanks Chee. $\endgroup$
    – Ali
    Commented May 21, 2020 at 11:12

0

Reset to default

You must log in to answer this question.