Timeline for Is there a canonical split signature metric on $\mathbb{P}^n\times\mathbb{P}^{n\,\ast}$?
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Oct 20, 2015 at 6:39 | vote | accept | Giovanni Moreno | ||
| Oct 20, 2015 at 6:23 | comment | added | Robert Bryant | @GiovanniMoreno: Because it's a symmetric space, the geodesics (null or not) can be computed explicitly via the exponential map in the group, in this case, $\mathrm{SL}(V)$, and I think that the behavior of the geodesics depends greatly on the relation of the two tangent components $P{\otimes}Q^* = \mathrm{Hom}(Q,P)$ and $Q{\otimes}P^* = \mathrm{Hom}(P,Q)$. Null is not a very strong condition, though, and a great variety of behaviors can exhibited by null geodesics when $p$ and $q$ are large. (When $p$ or $q$ are equal to $1$, it's much simpler, of course, but even there, there are 4 types.) | |
| Oct 19, 2015 at 16:55 | comment | added | Giovanni Moreno | I need time to figure out the complex case, but the rest is crystal-clear! Do you have some quick intuition/reference about (null) geodesics in the real case? I would expect that a pair of curves $(P(t),\pi(t))$ defines a null geodesic iff $\pi(t)$ is somehow related to (one of) the osculating space(s) to $P(t)$, but without familiarity with this stuff I can just guess. I suspect that some classical constructions like tangent, osculating, secant, dual varieties, can be recast in geodesic terms (or higher order conditions): I'm sure much has been done, but I don't know exactly what to look for! | |
| Oct 19, 2015 at 13:45 | history | edited | Robert Bryant | CC BY-SA 3.0 |
Fixed some typos and formatting errors
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| Oct 19, 2015 at 11:35 | history | answered | Robert Bryant | CC BY-SA 3.0 |