Let the Gauss-Bonnet form be $\Omega\propto\text{Pf}(\Omega^i{}_j)$ with $\Omega^i{}_j$ the curvature 2-form of an even-dimensional manifold with dim=$n$. The Gauss-Bonnet form is exact, as showed in the explicit construction in Chern 1944 (also available here). We may write $\Omega = d\Pi$ with $\Pi$ a rank $n-1$ form. Chern gave a construction for $\Pi$ in terms of an arbitrary vector field.

Compare this to the Chern-Simons form, as developed in Chern and Simons 1974. The Chern-Simons form is also exact, and Chern and Simons 1974 give an explicit construction without any arbitrary vector field—just in terms of the connection 1-form and the curvature 2-form.

Question: is there a modern version of Chern 1944 which, similar to Chern and Simons 1974, gives an explicit construction for $\Pi$ without reference to an arbitrary choice of vector field, but rather is only in terms of the connection 1-form and curvature 2-form?

Let the Gauss-Bonnet form be $\Omega\propto\text{Pf}(\Omega^i{}_j)$ with $\Omega^i{}_j$ the curvature 2-form of an even-dimensional manifold with dim=$n$. The Gauss-Bonnet form is exact, as shown in the explicit construction in Chern 1944 (also available here). We may write $\Omega = d\Pi$ with $\Pi$ a rank $n-1$ form. Chern gave a construction for $\Pi$ in terms of an arbitrary vector field.

Compare this to the Chern-Simons form, as developed in Chern and Simons 1974. The Chern-Simons form is also exact, and Chern and Simons 1974 give an explicit construction without any arbitrary vector field—just in terms of the connection 1-form and the curvature 2-form.

Question: is there a modern version of Chern 1944 which, similar to Chern and Simons 1974, gives an explicit construction for $\Pi$ without reference to an arbitrary choice of vector field, but rather is only in terms of the connection 1-form and curvature 2-form?