Hmmmmm Endiku...dressage is just that - riding without hands? *raises eyebrow*. Sorry mate, can't agree with you there.
When we talk about our 3 main/basic aids, we are talking the seat (so from your backside to your head, can also include upper leg depending on who's teaching you), the leg (calf, heel if required) and hand (shoulder to hand itself).
Ideally, and not only in dressage but in all english disciplines, we ask the horse to react first to our seat, then our leg, and THEN our hand. Because we always ask with the seat first, out seat becomes our strongest aid - and when you think about it, our seat is the biggest area of our body, so of course its going to have the most influence of the horse. To effectively use our seat, we NEED excellent core strength. Our seat is used to slow the horse, to collect the horse, to steer the horse... for everything.
So say we're asking for a walk to halt transition. The horse is actively walking forward into the bridle, and we want to ask him to halt. We first 'close' the seat. So stop your body moving with the motion of the walk. We then 'close' the leg. Closing the leg lightly onto the horse's sides encourages the hind legs to stay travelling so that when they stop, they come under the horse and 'sit' rather than being left behind and tipping the horse onto the forehand. A DOWNWARD TRANSITION IS STILL A FORWARD MOVEMENT. Hold your core/seat, and only if the horse does not respond by slowing/stopping, do you 'close' your rein. Closing the rein is not a pull back action though, you are simply taking up a little more weight in the rein. Do not allow to close to be a constant, so if the horse does not respond initially, add a little more weight to the rein and continue in this fashion until you get the halt.
If you allow the rein to be a constant through a transition like this, you then teach the horse to lean and become numb to the pressure.
In dressage, we always have a contact with rein. Riding with loops in the reins is in fact, despite popular and 'uneducated' belief, undesirable, as there is no connection to the forehand from the hindquarters. The rein provides the link to the bridle from the hind legs, and 'completes the circuit' of engagement and throughness. Hence, dressage is absolutely NOT 'riding without reins'.
As for turning with no reins. Try this. I want you to put your horse on a 10m circle using ONLY your seat and leg. Throw your reins up his neck for this exercise, no contact - just to prove a point that seat and legs are what influences the turn not rein, we are not asking for a connection in this exercise.
So walking on a straight line, actively forward. Say your want to make your circle at the letter B in the arena on the left rein. Turn to look at your horse's inside (left) hip and put your inside (left) leg on the girth. I'll put money on it that your horse will turn ;)
Now of course this is exaggerating the use of seat to perform a turn, but it shows that your body really does influence the horse significantly. When riding a 10m circle on my own horse, I simply ride my inside hip a little forward, slightly weight my inside stirrup, take my inside knee and thigh off the saddle while closing my outside knee and thigh onto the saddle, and turn my shoulders to match the turn of the circle. The horse follows and performs a dead straight, accurate 10m circle.
Now if I were to ride this 10m circle using my reins, say by pulling on my inside rein. As soon as you touch your inside rein, you immediately lose the inside hind leg. So you have already started your circle unbalanced and crooked. The horse's quarters are to the outside. Now when you lose the inside hind and pull on the inside rein, you're also going to tip the horse's weight onto its inside shoulder, and to compensate, it then has to bulge through the outside shoulder to balance. The neck becomes over bent to the inside... and you have made a real mess of a 10m circle.
It is the rider's responsibility to keep the horse upright and straight in every exercise that it performs. Whether a 20m circle in trot or a canter pirouette, the horse MUST be upright and straight in consideration of the exercises demands.
And keeping a horse upright and straight physically CANNOT come from the rider's hands, it MUST come from the riders seat and a little of the leg.
The rein just cannot, on its own, prevent the quarters from swinging, the shoulders from dropping or bulging, and maintain the bend. Physics, the rein just can't do that! It is merely a supporting aid, one which completes the circuit of connection.