For an object in gravitational field, the gravitational potential...

The work done by the gravitational force on an object of mass $m$ in moving it from point $A$ to point $B$ is given by $W = -\Delta U = -m(V_B - V_A) = m(V_A - V_B)$, where $V_A$ and $V_B$ are the gravitational potentials at points $A$ and $B$ respectively.

It is given that the gravitational potential is the same at points $A$ and $B$, which means $V_A = V_B$.

Substituting this into the work equation, we get $W = m(V_A - V_A) = 0$.

The difference in the gravitational field at the two points does not affect the total work done between these equipotential points, as gravity is a conservative force and work done is independent of the path taken.

Therefore, no work is done by gravity in moving the object from point $A$ to point $B$.