With reference to the questions Does the Riemann-Christoffel curvature determine the connection? and When is a given matrix of two forms a curvature form?, and recalling the following important result stated in Theorem 7 in "J C Talvacchia. Prescribing the curvature of a principal bundle connection. University of Pennsylvania, 1989. PhD thesis, Dissertations available from ProQuest. Paper AAI9004831. http://repository.upenn.edu/dissertations/AAI9004831.":
"Let $P$ be an $SL(3,\mathbb{R})$-bundle over a three dimensional base manifold. Then a generic analytic $sl(3,\mathbb{R})$-valued two form is locally the curvature form of connection on $P$.",
I would like to ask if there is any extension of this particular result (i.e., existence of a connection over an $SL(3,\mathbb{R})$-bundle with three dimensional base manifold whose curvature form is a given $sl(3,\mathbb{R})$-valued two form) beyond real analytic category?
Thanks in advance!
