Why is there no current in this part of the circuit?
Assuming ideal circuit elements and conductors:

Simply because there is no return path for \$i_o\$.

The entire lower line (call it ground or common) in the schematic is considered to be one node, therefore there is considered to be no voltage difference across it either.

(Suppose that there was a current flowing in \$i_o\$. Then the only way back to the starting point would be trough the same node but in opposite direction, meaning that the sum of the currents is \$i_o + (- i_o) = 0\$.)