Induced Field Theory on the Brane World - Gravity, Extrinsic Curvature,   and Gauge Fields
  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. 
Keiichi Akama, Takashi Hattori