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Let $M$ be a $4m$-dimensional Quaternion-Kähler manifold of positive scalar curvature. Does there exist an $n$ large enough, so that $M$ can be embedded inside $\mathbb{H}P^n$ via a quaternionic embedding (see my remark below)? If not, can one formulate some necessary and sufficient conditions for such an $n$ and such an embedding to exist?

Remark: there is a theory of quaternionic mappings in an old paper by Galicki for instance, if I remember correctly, but I cannot seem to find it.

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    $\begingroup$ The answer is 'no', in general. In fact, as is easy to show, the only 'quaternionic' submanifolds of $\mathbb{HP}^n$ are the 'linear' submanifolds $\mathbb{HP}^m\subset\mathbb{HP}^n$. $\endgroup$
    – Robert Bryant
    Aug 5, 2017 at 2:46
  • $\begingroup$ Ah ok. Thank you Prof. Bryant. It looks like quaternionic maps are scarse. $\endgroup$
    – Malkoun
    Aug 5, 2017 at 7:00
  • $\begingroup$ Heck, Quaternion-Kaehler manifolds are scarce: I don't think we know of any other than the Wolf manifolds... $\endgroup$
    – W. Cadegan-Schlieper
    Aug 5, 2017 at 7:25
  • $\begingroup$ I was off on a wrong track again... My intuition is that $\mathbb{H}P^m$ are kind of "universal" among QK manifolds (maybe restricted to have positive scalar curvature). Maybe similar to how $SL(2,\mathbb{C})$'s are "universal" in the theory of complex semisimple Lie groups, in some sense. Perhaps the only way to formalize this intuition is via the LeBrun-Salamon conjecture. $\endgroup$
    – Malkoun
    Aug 5, 2017 at 9:35
  • $\begingroup$ What I mean is that many constructions that work for $\mathbb{H}P^m$, and are "natural" enough, seem to extend to Quaternion-Kähler manifolds in general (with maybe some restriction on the scalar curvature). Perhaps $\mathbb{H}P^m$ can be replaced by a family of Wolf spaces associated to a family of classical groups. I am trying to formalize this vague intuition. I have been unsuccessful so far. $\endgroup$
    – Malkoun
    Aug 5, 2017 at 9:39

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