A few notes on induction motors, and VFDs, might clarify a few characteristics.
The plate on an induction motor will have a rated voltage and a rated current. In the UK these will normally be specified at 50Hz. The rated voltage is usually the supply voltage, nominally 380VAC for 3-phase in the UK. If the rated current is drawn at the rated voltage this gives the rated power of the motor. Since the speed is fixed by the 50Hz supply the rated torque can be calculated. The theoretical synchronous speed of the motor, known as base speed, is set by the supply frequency and number of poles.
Now consider a VFD driving the motor at a variable frequency, and hence variable speed. First assume that the frequency increases above 50Hz. By definition the rated current is drawn at the rated voltage at 50Hz. As the frequency increases more voltage is needed to maintain the rated current. But a higher voltage is not possible. So the voltage stays constant and as the frequency increases the current drops. To a first approximation motor torque is proportional to current, so the torque drops too. To summarise, as the frequency rises the current and torque fall in proportion. But since the speed has increased, and power is torque times angular velocity (aka speed), the drop in torque is cancelled out by the increase in speed and the power is constant. So above base speed the motor is constant power, but not constant torque.
Now consider the case where the frequency drops below 50Hz. If the applied voltage is constant the current will rise above the rated current and the windings may well overheat due to I2R. To keep the current at the rated value the voltage is reduced, hence the V/F characteristic (also known as scalar control) described by Clive. Since the current is kept at the rated current the torque stays constant. But since the speed has decreased in proportion to the frequency the power decreases. Below base speed the motor is constant torque but not constant power.
Also alluded to by Clive more complex VFDs use vector control, or field-oriented control. The use of vector control allows more control of the motor torque characteristics compared to scalar control. The mathematics of vector control is fairly complex and involves transforming the 3-phase rotating field to a stationary two dimensional space with two orthogonal currents, which can be represented by a vector, hence the name. One current controls magnetic flux and the other controls torque. These are then transformed back to a rotating field to control the currents in the motor winding.
Interesting fact: The transform between rotating field and stationary vector is called the Clarke transform, after Edith Clarke the first professional female electrical engineer in the US.
Julie