Timeline for Geometry of the complex quadric

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Nov 5, 2016 at 7:43	comment	added	alvarezpaiva		Thanks @RobertBryant. Yes, I'm interested in the quadric as a homogeneous space of the orthogonal complex group and specially about the geometry of real and complex curves (the method I'm working on provides a way to study those, but I would like to compare to "classic" work). The cases n = 2, 3, 4, and 5 should suffice for me to get an idea of how the method works in this case. Do you have any references?
Oct 30, 2016 at 1:02	comment	added	Robert Bryant		Do you want to know about real curves, surfaces, etc. or do you want to know about complex curves and surfaces, etc? I'm assuming that you want to consider $Q_{n-1}$ as a homogeneous space of $\mathrm{O}(n{+}1,\mathbb{C})$, not as a Riemannian space. The case $n=3$ is not so interesting because $Q_2 = \mathbb{CP}^1\times\mathbb{CP}^1$. The case $n=4$ was treated classically, taking advantage of the classical isomorphism $B_2 = C_2$ and the case $n=5$ has a similar (but not as developed) story using $D_3 = A_3$. Above that, I think not much is known in detail.
Oct 25, 2016 at 23:58	answer	added	314159.		timeline score: 3
Oct 25, 2016 at 19:29	history	edited	alvarezpaiva	CC BY-SA 3.0	added 5 characters in body
Oct 25, 2016 at 19:15	history	asked	alvarezpaiva	CC BY-SA 3.0