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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 $$(x,y,z)\mapsto \alpha\cdot(\beta+x^2,\beta+y^2,\beta+z^2,\sqrt{2}xy,\sqrt{2}yz,\sqrt{2}zx)$$ 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 $\mathbb{R}^{N+1}$.

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 $\mathbb{R}^{N+1}$.

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 embedding of $\mathbb{R}^{n+1}/\iota$ 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 $$(x,y,z)\mapsto \alpha\cdot(\beta+x^2,\beta+y^2,\beta+z^2,\sqrt{2}xy,\sqrt{2}yz,\sqrt{2}zx)$$ 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 $\mathbb{R}^{N+1}$.

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 $$(x,y,z)\mapsto \alpha\cdot(\beta+x^2,\beta+y^2,\beta+z^2,\sqrt{2}xy,\sqrt{2}yz,\sqrt{2}zx)$$ 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 $\mathbb{R}^{N+1}$.

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 embedding of $\mathbb{R}^{n+1}/\iota$ 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 $$(x,y,z)\mapsto \alpha\cdot(\beta+x^2,\beta+y^2,\beta+z^2,\sqrt{2}xy,\sqrt{2}yz,\sqrt{2}zx)$$ for approprately chosen $\alpha$ and $\beta$.)

So, "yes", for any centrally symmetric surface $\Sigma$ in $\mathbb{R}$, the quotient $\Sigma/\iota$ admits an explicit embedding in $\mathbb{R}^{N+1}$.

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 embedding of $\mathbb{R}^{n+1}/\iota$ 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 $$(x,y,z)\mapsto \alpha\cdot(\beta+x^2,\beta+y^2,\beta+z^2,\sqrt{2}xy,\sqrt{2}yz,\sqrt{2}zx)$$ 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 $\mathbb{R}^{N+1}$.