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For a smooth, real surface $\Sigma$, its bundle of symmetric, bi-linear forms $S^2T\Sigma$ reduced to a $PGL(2,\mathbb{R})$ structure. A similar reduction(with different structure group) can be done for other tensor bundles.

  1. Where can I find a good reference on this type of reductions?

  2. $PGL(2,\mathbb{R})$ acts by conjugation on the space of square matrices $M(3,\mathbb{R}).$ Can one find any invariant polynomials associated to this action?

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The group $GL(n,\mathbb{R})$ does not act by conjugation on symmetric bilinear forms. If $A$ is the symmetric $n\times n$ matrix describing one such form in a given basis and $S$ is a linear invertible operator defining a basis change, then in the new basis the matrix is $SAS^t$. It is not clear to me how $PGL$ acts on the space of symmetric forms.

As for the invariants of the action $PGL(2)$ on $M(2)$ they're the same of the action of $GL(2)$ on $M(2)$. You can decide this using the theory of Jordan normal forms.

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    $\begingroup$ Thank you Liviu. I did not say that $GL$ acts by conjugation on symmetric bilinear forms. $GL(3)$ acts by matrix multiplication on the frames of the bundle $S^2T\Sigma$. The subgroup of $GL(3,)$ defined as $$ \left( \begin{array}{lll} a^2 & 2ab &b^2 \\ ac & bc+ad & bd \\ c^2 & 2cd & d^2 \end{array} \right) $$ is isomorphic to $PGL(2,\mathbb{R}).$ $\endgroup$
    – Mike Cocos
    Jan 8, 2016 at 16:44
  • $\begingroup$ @MikeCocos Got it $\endgroup$
    – Liviu Nicolaescu
    Jan 8, 2016 at 17:09

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