Timeline for Isometric imbedding of ellipsoidal projective plane

8 events

when toggle format	what		by	license	comment
Sep 16, 2017 at 0:24	history	edited	Anton Petrunin	CC BY-SA 3.0	added 42 characters in body
Aug 29, 2017 at 16:37	comment	added	Anton Petrunin		@RobertBryant The map with $\alpha=1$ and $\beta=0$ is expanding, so moving the center away and rescaling we get a length preserving map.
Aug 29, 2017 at 16:31	comment	added	Robert Bryant		You are very kind. Mainly, I just wanted to convince myself that the roots were real, which was not obvious to me. I suppose it's also worth mentioning that, for the analogous construction for general $n$, the solutions have $\alpha^2= 1/2$ and $\beta = (-1\pm\sqrt{n{+}2})/(n{+}1)$.
Aug 29, 2017 at 16:19	history	edited	Anton Petrunin	CC BY-SA 3.0	added 29 characters in body
Aug 29, 2017 at 15:08	history	edited	Robert Bryant	CC BY-SA 3.0	Added a missing exponent
Aug 29, 2017 at 12:37	comment	added	Robert Bryant		I believe that $\alpha = 1/\sqrt2$ and either $\beta=-1$ or $\beta = \frac13$ will give the desired isometric embedding of $\mathbb{RP}^2$ into $S^5\subset\mathbb{R}^6$. It may be worth mentioning that this map is equivariant with respect to actions of $\mathrm{SO}(3)$ on $\mathbb{R}^3$ and $\mathbb{R}^6$.
Aug 28, 2017 at 23:07	history	edited	Anton Petrunin	CC BY-SA 3.0	added 266 characters in body
Aug 28, 2017 at 21:11	history	answered	Anton Petrunin	CC BY-SA 3.0