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 $~

  • $\begingroup$ Out of interest, what is the proof that $S^3/Q_8$ is the complete flag manifold of $\mathbb{R}^3$? $\endgroup$
    – Nick L
    Dec 21, 2021 at 22:50
  • 3
    $\begingroup$ @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'. $\endgroup$
    – mme
    Dec 21, 2021 at 23:09
  • $\begingroup$ @NickL: I wrote up an argument of Kusner's, here ldtopology.wordpress.com/2012/07/12/… $\endgroup$ Jan 17, 2022 at 5:55
  • $\begingroup$ @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! $\endgroup$ Jan 19, 2022 at 18:15

1 Answer 1


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.


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