No torque at synchronous speed

The statement is true because an induction motor works by induction, and induction needs relative motion between the rotating magnetic field and the rotor conductors.

1. At synchronous speed $N_s = \frac{120f}{P}$, the rotor and the field rotate together. The slip is zero:

$$s = \frac{N_s - N_r}{N_s} = 0$$

2. The rotor conductors do not cut any flux, so the rotor emf is $E_{2s} = sE_2 = 0$.
3. With no emf, the rotor current is $I_2 = 0$, so there is no rotor mmf and no force on rotor conductors.
4. Torque is $T = \frac{3}{\omega_s}\cdot\frac{sE_2^2R_2}{R_2^2+(sX_2)^2}$, which is zero at $s = 0$.
5. Without torque, friction and windage slow the rotor. Slip then appears, an emf is induced, and torque is produced again.

So the motor can only run at a speed slightly below synchronous speed, where the rotor current produces just enough torque to supply the load and the mechanical losses. At no load, the slip is very small (about 0.5 %) but not zero.

This also shows on the T-S curve, where the torque curve passes through zero at $s = 0$.