Robert Bryant's answer to [Isometric embedding of SO(3) into an euclidean space](https://mathoverflow.net/questions/161295/isometric-embedding-of-so3-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 $~



Embedding round manifolds into low dimensional spheres
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.


