# Embedding round manifolds into low dimensional spheres

Robert Bryant's answer to Isometric embedding of SO(3) into an euclidean space mentions that there is an isometric embedding of the round tetrahedral space $$SO_3/A_4$$ into the round sphere $$S^6$$.

I like that fact. Dreaming along those lines, here are some more facts: (and I am interested in any similar examples that MO knows about)

-$$S^4$$ contains isometrically embedded copies of the the round projective plane $$\mathbb{R}P^2$$ and the round prism manifold $$S^3/Q_8$$ (this happens to be the manifold of complete flags in $$\mathbb{R}^3$$) where $$Q_8$$ is the eight element quaternion group

-This doesn't quite fit the theme of roundness but $$S^5$$ contains a weird distorted (not round but still Riemannian homogeneous, in particular with isometry group $$SU_2$$) copy of $$\mathbb{R}P^3$$ . This can be seen as the unit tangent bundle of the sphere $$S^2$$ inside $$T(\mathbb{R}^3)=\mathbb{R}^6$$. This weird $$\mathbb{R}P^3$$ is isometrically double covered by the Berger sphere $$S^3$$.

-$$S^6$$ contains an isometrically embedded round tetrahedral space $$SO_3/A_4$$ as previously mentioned

-$$S^8$$ contains a round $$\mathbb{R}P^3$$

-$$S^N$$ contains a round projective space $$\mathbb{R}P^n$$ where $$N=\frac{(n+1)(n+2)}{2}-2$$, moreover $$N$$ is the minimal such dimension.

Moreover all these embeddings are equivariant with respect to the action of the Isometry group (which is transitive-- these manifolds are all Riemannian homogeneous in addition to being round).

I am aware that for sufficiently large $$N$$ every Riemannian manifold isometrically embeds in round $$S^N$$. What I am really looking for are (equivariant) isometric embeddings of round (homogeneous) Riemannian manifolds into low dimensional round spheres. The properties in parentheses are especially interesting.

Disclaimer: ~everything is smooth I know Nash-Kuiper does very low dimensions for $$C^1$$~

• Out of interest, what is the proof that $S^3/Q_8$ is the complete flag manifold of $\mathbb{R}^3$? Dec 21, 2021 at 22:50
• @NickL $S^3/Q_8 = SO(3)/S(O(1) \times O(1) \times O(1))$ ($SO(3)$ mod the subgroup of diagonal matrices), because the preimage of this subgroup under the projection $\pi: S^3 \to SO(3)$ is precisely $Q_8$. Now the key is that $A \in SO(3)$ gives rise to a full flag as $(A\text{span}(e_1), A\text{span}(e_1, e_2), \Bbb R^3)$. This is a surjective map, which is 4 : 1; you get the same full flag from $A$ and $A'$ iff the columns of A are +- the columns of A'.
– mme
Dec 21, 2021 at 23:09
• @NickL: I wrote up an argument of Kusner's, here ldtopology.wordpress.com/2012/07/12/… Jan 17, 2022 at 5:55
• @RyanBudney do you happen to know what the story is for decomposing $S^6$ this way using the irrep of $SO_3$ on $R^7$? Or for decomposing $S^8$ this way using the conjugation action of $SO_4$ on $R^9$ the space of traceless symmetric $4 \times 4$ matrices? Also yes I read that exact write up months ago that's how I first heard about a lot of this stuff! Crazy actually meeting you now on MO! Jan 19, 2022 at 18:15

Here are a couple more examples I whipped up with the help of MSE/MO. I think they are enough to constitute an answer.

As I mentioned in my question Robert Bryant inspired me with his example of

• The tetrahedral space $$SO_3/A_4$$ as the orbit of $$xyz$$ with respect to the representation of $$SO_3$$ on the seven dimensional space of harmonic homogeneous polynomials of degree 3 in three real variables $$x,y,z$$. Since the representation is orthogonal this gives an isometric embedding into the round sphere $$S^6$$ inside of this seven dimensional real representation.

So the other stuff I cooked up is all the other Riemannian homogeneous spherical 3 manifolds

• The homogenous lens spaces $$L_{n,1}$$ as the stabilizer of $$u^n$$ in representation of $$SU_2$$ on the space of homogeneous polynomials of degree $$n$$ in two complex variables $$u,v$$. This representation is complex dimension $$n+1$$ so this isometrically embeds the lens space into the round sphere $$S^{2n+1}$$

• for the even lens spaces with $$n \geq 3$$ $$L_{2n,1} \cong SU_2/C_{2n} \cong SO_3(\mathbb{R})/C_n$$ we can do much better by realizing them as the orbit with respect to the representation of $$SO_3$$ on the $$2n+1$$ dimensional real vector space of degree $$n$$ homogeneous harmonic polynomials in three real variables $$x,y,z$$ of the polynomial $$Re[(x+iy)^n]+2Im[(x+iy)^n]$$ here $$2$$ can be replaced by any real number other than $$0$$ or $$\pm 1$$.

• The homogeneous prism manifolds as the orbit with respect to the representation of $$SO_3$$ on the $$2n+1$$ dimensional real vector space of degree $$n$$ homogeneous harmonic polynomials in three real variables $$x,y,z$$ of the polynomial $$Re[(x+iy)^n]= \sum_{k=0}^D (-1)^k {n \choose 2k} x^{n-2k}y^{2k}$$ where $$D$$ is the floor of $$n/2$$ (so $$n/2$$ for even case and $$(n-1)/2$$ for odd case. Note that this polynomial is harmonic because it is the real part of the holomorphic function $$w^n=(x+iy)^n$$. Since this is a $$2n+1$$ real dimensional orthogonal representation this yields an isometric embedding of round prism manifold $$SO_3/D_{2n}$$ into the round sphere $$S^{2n}$$

• The round octahedral space $$SO_3/S_4$$ embeds isometrically in $$S^8$$ as the orbit of the harmonic polynomial $$x^4+y^4+z^4-3y^2z^2-3x^2z^2-3x^2y^2$$

• round icosahedral space/ Poincare homology sphere embeds isometrically in $$S^{12}$$ as the orbit of the harmonic polynomial $$21(2-\sqrt{5})(x^2-\phi^2y^2)(y^2-\phi^2z^2)(z^2-\phi^2x^2)+(x^2+y^2+z^2)^3$$ where $$\phi$$ is the golden ratio.

Indeed I claim these dimensions

• $$d=2n+1$$ for the lens spaces $$L_{2n,1}\cong SO_3/C_n$$ ( $$n \geq 3$$)

• $$d=2n+1$$ for the prism spaces $$SO_3/D_{2n}$$ ($$n \geq 2$$)

• $$d=7$$ for the tetrahedral space $$SO_3/A_4$$

• $$d=9$$ for the octahedral space $$SO_3/S_4$$

• $$d=13$$ for the icosahedral space $$SO_3/A_5$$ (note this this almost saturates the bound from Gunther's version of the Nash Embedding Theorem of $$d=14$$ for isometrically embedding a generic compact 3 manifold )

are minimal for isometrically equivariantly embedding these (round) spherical 3 manifolds in Euclidean space $$\mathbb{R}^d$$. And moreover that the final three dimensions are minimal even among all isometric ( not necessarily equivariant) embeddings.