  We present the effective gravitational field equations in a 3-brane world
with Euler-Poincare term and a cosmological constant in the bulk spacetime. The
similar equations on a 3-brane with $\mathbb{Z}_2$ symmetry embedded in a five
dimensional bulk spacetime were obtained earlier by Maeda and Torii using the
Gauss-Coddazzi projective approach in the framework of the Gaussian normal
coordinates. We recover these equations on the brane in terms of differential
forms and using a more general coordinate setting in the spirit of Arnowitt,
Deser and Misner (ADM). The latter allows for acceleration of the normals to
the brane surface through the lapse function and the shift vector. We show that
the gravitational effects of the bulk space are transmitted to the brane
through the projected ``electric'' 1-form field constructed from the conformal
Weyl curvature 2-form of the bulk space. We also derive the evolution equations
into the bulk space for the electric 1-form field, as well as for the
``magnetic'' 2-form field part of the bulk Weyl curvature 2-form. As expected,
unlike on-brane equations, the evolution equations involve terms determined by
the nonvanishing acceleration of the normals in the ADM-type slicing of
spacetime.
