I recently came across the notion of a polynomial, resp. metallic manifold. Since I have never heard of it before and I assume that the same applies to most of you, I wanted to share its definition with you.

Let \(M\) be a (smooth) manifold. A polynomial structure on \(M\) is a \((1,1)\)-tensor field \(\Phi\) on \(M\) satisfying a polynomial relation \[\Phi^n + a_{n-1}\Phi^{n-1} + \cdots + a_1 \Phi + a_0 \mathrm{Id} = 0.\]

For example, every almost complex structure \(J\) is a polynomial structure since it satisfies the relation \(J^2 + \mathrm{Id} = 0\).

A special type of polynomial structures are metallic structures: These are \((1,1)\)-tensor fields \(\Phi\) on \(M\) satisfying a relation of the form \[\Phi^2 – p \Phi – q \mathrm{Id} = 0,\] where \(p\) and \(q\) are positive integers. The name comes from the golden ration \(\phi\) which satisfies the equation \(\phi^2-\phi-1=0\).

Ok, so I resp. we know now what polynomial and metallic structures are. What do we do with them? And here comes the problem: I just don’t know! I have looked at three papers dealing with them (a survey, the initial paper on polynomial structures, the initial paper on metallic structures) and none of them gave me reasons to be concerned with these structures. They all prove only results internal to the theory, and there do not seem (at least to my eye) to be important/interesting examples of such structures where these internal results yield something new.

So, to everybody working with these structures and reading this: Enlighten me! Why are you interested in these structures?