Timeline for Lower bound for domain of exponential map on Lorentzian manifolds
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 19, 2020 at 18:28 | comment | added | user143410 | Right, I wouldn't expect a uniform bound and, as your example indicates, I should have anticipated that global properties would enter. Essentially, the (ball-shaped) domain of the exponential map is determined by the length of the shortest inextendible geodesic passing through $p$. Perhaps proofs for classic singularity theorems will provide some insight for me. I would think there must be a way to estimate the size of $r_0$ for say a spacetime ($R^D$,$g_{ab}$) with $g_{ab}$ non-flat and satisfying some (physically-reasonable) global constraints. | |
| Apr 18, 2020 at 17:43 | comment | added | Igor Khavkine | @user143410, consider the example of the interior of a Euclidean half-space. The $r_0$ at $p$ cannot be greater than the distance of $p$ to the (open) boundary of the half space. This means that there is no uniform lower bound for $r_0$ on the manifold. Also, whatever the lower bound is at $p$, you cannot get it from local curvature, since the manifold is everywhere flat. The lower bounds that you are looking for somehow depend on the structure of this space "at large". If this example is not what you had in mind, you need some conditions to eliminate it. | |
| Apr 18, 2020 at 16:04 | comment | added | user143410 | Apologies for any confusion due to imprecise phrasing. However, I'm glad you mention the Chen-LeFloch paper, Igor. Note their injectivity radius bound (Thm 1.1) assumes the exponential map is defined on ball $B_T(r_0)$ for some $r_0>0$. Their bound is then stated in terms of $r_0$. Hence, if one had a lower bound on the (ball-shaped) domain of the exponential map (i.e., a lower bound on $r_0$), Chen-Lefloch would then provide a bound on the minimum possible injectivity radius (for some fixed metric with bounded curvature). So, a bound on $r_0$--which is what I want--would then be very useful. | |
| Apr 18, 2020 at 11:44 | comment | added | Igor Khavkine | I grant that your interpretation is a better literal reading of the OP, but I think there is some liberty in interpreting what "well defined" means. I suspect that I answered the intended question, but I may be wrong. | |
| Apr 18, 2020 at 9:10 | comment | added | Gro-Tsen | I believe there's some confusion. I think the injectivity radius is the largest $r$ such that $\exp_p$ is injective on $B_T(r)$, but this may be smaller than the largest $r$ such that $\exp_p$ is defined, which OP was asking about: on the unit sphere, as on any geodesically complete manifold, $\exp_p$ is defined on the entire tangent space $T_p M$, but it is injective only up to $r = \pi$. | |
| Apr 18, 2020 at 7:45 | history | answered | Igor Khavkine | CC BY-SA 4.0 |