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The famous Whitney immersion theorem states that any real projective space $\mathbb{RP}^n$ can be immersed into $\mathbb{R}^{2n}$.

However, I haven't found information about the submersion corresponding to the theorem.

Of course, a compact manifold can not be submerged into Euclidean space. So let's consider $\mathbb{RP}^n \times (-\epsilon,\epsilon)$ for a positive number $\epsilon$.

For what combination of $n\geq m$ can $\mathbb{RP}^n \times (-\epsilon,\epsilon)$ be submerged into Euclidean space $\mathbb{R}^m$?

Here are my observations so far:

  1. $f:S^1\times (-\epsilon, \epsilon)\hookrightarrow \mathbb{R}^2 - \{O\} \rightarrow \mathbb{R}^2$ given by $f:(x_1, x_2)\mapsto (x_1^2-x_2^2,2 x_1x_2)$ induces a submersion $\mathbb{RP}^1\times (-\epsilon,\epsilon)\rightarrow \mathbb{R}^2$.

  2. A similar construction used in Hopf fibration gives a submersion: $f:\mathbb{R}^4- \{O\} \rightarrow \mathbb{R}^3\approx\mathbb{C}\times \mathbb{R}$ given by $f:(x_1, x_2, x_3, x_4)\mapsto (z_1z_2^*, |z_1|^2-|z_2|^2)$, where $z_1 = x_1+ix_2$ and $z_2 =x_3+ix_4$ gives a submersion $\mathbb{RP}^3\times (-\epsilon,\epsilon)\rightarrow \mathbb{R}^3$.

  3. Similarly, Hopf fibrations using quaternions and octonions give submersions $\mathbb{RP}^7\times (-\epsilon,\epsilon)\rightarrow \mathbb{R}^5$ and $\mathbb{RP}^{15}\times (-\epsilon,\epsilon)\rightarrow \mathbb{R}^9$.

I think we could find some submersions in a smaller codimension.

Does anyone know the answer or have a proper reference?

Thank you in advance for your help.

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    $\begingroup$ One should be able yo get an answer from Antony Phillips' theorem, an h-principle for submersions. $\endgroup$
    – Moishe Kohan
    Commented Feb 5 at 4:32
  • $\begingroup$ Often there are restrictions, for example you can't do it for $n=2$ and $m=3$ since the domain is non-orientable. This argument works for all $n$ even and $m=n+1$. $\endgroup$
    – Ryan Budney
    Commented Feb 5 at 4:43
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    $\begingroup$ All the maps in your example have differential $0$ at $0$, so they are not submersions. $\endgroup$
    – Sebastian Goette
    Commented Feb 5 at 7:37
  • $\begingroup$ @MoisheKohan Thank you for the reference! So, as far as I understand, there exists a submersion if and only if there exists a section in $T_pM$ where $T_pM$ is the bundle of $p$-frames tangent to $M$. $\endgroup$
    – GHG
    Commented Feb 6 at 1:28
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    $\begingroup$ Right, where $p$ is the dimension of the target Euclidean space (your notation is $m$). Then the question becomes a matter of the obstruction theory. But make sure you never use the notation $T_pM$ for this bundle. $\endgroup$
    – Moishe Kohan
    Commented Feb 6 at 1:35

1 Answer 1

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By Theorem B of 1, there exists a submersion $M\rightarrow \mathbb{R}^m$ if and only if there exists a section in $F_m(M)$ where $F_m(M)$ is the bundle of $m$-frames tangent to $M$.

It is equivalent to the existence of $m$ independent vector fields.

By Theorem 1.1 of 2, the maximum number of linearly independent vector fields on $\mathbb{RP}^n$ equals $8a+2^b−1$ where $n+1=2^{4a+b}×c$ for $0≤b≤3$ and an odd number $c$.

Therefore, $\mathbb{RP}^n\times(−ϵ,ϵ)$ has $8a+2^b$ independent vector fields and thus can be submerged into $\mathbb{R}^{8a+2^b}$. This result coincides with Hopf fibration cases $(n,m)=(15,9),(7,8>5),(3,4>3)$ and $(2,2)$, also coincides with Ryan's observation on even $n$ cases.

Thank you again to Moishe Kohan and all others for your comments!

$[1]$ Phillips, Anthony. "Submersions of open manifolds." Topology 6.2 (1967): 171-206.

$[2]$ Davis, Donald. "Vector fields on $𝑅𝑃^{𝑚}× 𝑅𝑃ⁿ$." Proceedings of the American Mathematical Society 140.12 (2012): 4381-4388.

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  • $\begingroup$ Do you know if the maximum number of independent vector fields on $\mathbb{RP}^n\times(-\varepsilon, \varepsilon)$ is $8a+2^b$? In the case of spheres, the maximum number of independent vector fields on $S^n$ is $8a+2^b-1$, but $S^n\times(-\varepsilon, \varepsilon)$ admits $n + 1$ independent vector fields. It is known that $\mathbb{RP}^n$ is stably parallelisable iff $n = 1, 3, 7$ (in which case it is actually parallelisable) - see Theorem 3.1 of Korbaš - On Parallelizability and Span of the Dold Manifolds for example. $\endgroup$
    – Michael Albanese
    Commented Feb 7 at 12:56

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