  We show how the gravity, extrinsic curvature, and gauge field theories are
induced on dynamically localized brane world. They should obey the
Gauss-Codazzi-Ricci equation in addition to their own equations of motion. As
an example, we derive the solitonic solution for curved domain wall in five
dimensions in terms of gravity and extrinsic curvature fields of the brane, and
then derive the effective action for their field theory on the brane.
