TITLE: Isometric imbedding of ellipsoidal projective plane
QUESTION [9 upvotes]: Identifying antipodal points of an ellipsoid (with axes of different length) defines a Riemannian metric on the real projective plane $\mathbb RP^2$. Is there an explicit global isometric imbedding of this metric into Euclidean space $\mathbb R^N$? By explicit I mean using special functions, ellipsoidal harmonics, etc. The ambient dimension $N$ need not be 5 but perhaps not too large. Of course, the same question can be asked for any $\mathbb RP^n$ with an ellipsoidal metric.

REPLY [6 votes]: The Veronese embedding provides an isometric embedding $\mathbb{R}\mathrm{P}^n$ into $\mathbb{S}^N\subset \mathbb{R}^{N+1}$.
Taking the cone over this map produce an isometric embedding of the quotient space $\mathbb{R}^{n+1}/\iota$ into $\mathbb{R}^{N+1}$, where $\iota$ is the central symmetry $\iota\colon x\mapsto -x$.
(If $n=2$, then the needed length-preserving map $\mathbb{S}^2\to \mathbb{R}^6$ is defined by 
$$(x,y,z)\mapsto \alpha\cdot(\beta+x^2,\beta+y^2,\beta+z^2,\sqrt{2}{\cdot}x{\cdot}y,\sqrt{2}{\cdot}y{\cdot}z,\sqrt{2}{\cdot}z{\cdot}x)$$
for approprately chosen $\alpha$ and $\beta$.)
So, "yes", for any centrally symmetric surface $\Sigma$ in $\mathbb{R}^n$, the quotient $\Sigma/\iota$ admits an explicit embedding in $\mathbb{R}^{N+1}$.