  By considering the p-brane motion in G/K symmetric space bulk we identify the
G-invariant bulk metric in the solvable lie algebra gauge of the brane action.
After calculating the Levi-Civita connection of this bulk metric we use it in
the Gauss equation to compute the braneworld curvature in terms of the bulk
coordinates. Finally, by making use of the Gauss equation in the braneworld
Einstein equation we present a geometrical method of implementing the first
fundamental form in the gravitating brane dynamics for the specially chosen
symmetric space bulk case leading to an Einstein equation solely expressed in
terms of the bulk coordinates of the braneworld.
