The bow and the back crunch are two stretching exercises that stretch the horses muscles both in the topline and in the underline. Because both frontfeet and hindfeet do the same thing and stand at the same distance all the time, it's an even, total stretch; there's no left or right stretch.
Stretching is the same in horses and people; when you get out of bed and are asked to run 100 metres, you will do better (and less painfull!) if you stretch before. You'll become more supple because as your muscles are stretched out, they find it easier to contract again. Therefore it's not a bad idea to teach the bow and back crunch before or at the same time as you're teaching higher levels of collection like the piaffe. Especially the back crunch is of a lot of importance for that, as in collection you fold all the hipjoints together in small, sharp angles all the time. If you have a way to stretch them out again regularly too, your horse will become less tired and pained by it. If you ask for a collected movement, then a back crunch and then the collected movement again, you will see that the second is better than the first: the horses' muscles are free to contract again. Muscles get stronger by shortening and stretching repeatedly, than by staying contracted for a longer period of time. So stretches actually help get the muscles stronger even though the exercises themselves don't collect the horse.The bow (front crunch)
The bow is an exercise in which the horse lowers his shoulders to the ground, with his frontlegs stretched forward near horizontal, and the hindlegs standing straight under the body. Because the frontlegs are stretched out like that, the shoulderjoints all open up very wide: the shoulderblade rotates upwards to nearly horizontal, the angle with the upper arm bone (humerus) gets bigger and the angle between the humerus and the ulna too. Because this is such a mighty stretch, it can take a while before the horse performs it really low to the ground - and also if he is stiff on one side, his bow will become less deep. And because of this opening up of the entire shoulder, the horse will achieve more freedom in his frontlegs when moving - and will get a better Spanish walk, extended trot or passage for example.
With some trainers you see that they let the horse stand stretched out first before they ask the horse to bow, but that means that the hindlegs stay stretched behind the body too, while they should at least take over some of the bodyweight in order to not overload the frontquarters when they stretch down and out. You can teach your horse the bow by asking him to follow an(empty!) hand that slowly moves towards and between his frontlegs - and and reward in a normal position again when he does! This way he does stretch his frontquarters and frontlegs out, but his hindlegs remain engaged under the body -and still bear the bodyweight. At first you still ask him to touch your hand, then you can add the cue for this exercise by - for example - patting with your hand on his rump, touching there with a whip or merely gesturing towards the ground to ask him a bow. Over time you then not only to ask him deeper and deeper, but also to stay longer in the bow before you ask him to get up again.Back crunch
In the back crunch the horse essentially lowers his pelvis to the ground, with his hindlegs stretched out behind the body and the frontlegs standing straight under the shoulder. Because the hindlegs stretch out behind the body, the joints in the hindquarters between pelvis, humerus and tibia open up and the muscles that are tying those together stretch open. It's a really good way of stretching the horse between and before collected exercises, because in real collection on the spot, like in piaffe, pesade and levade, the horse not only moves his hindlegs further forward (tying the hindquarter joints together in strong, sharp angles) but also rotats the tail-end of the pelvis downward-and forward by shortening muscles in the belly. Only with a movement back and out of both the hindlegs at the same time like in back crunch or a fast gallop you stretch that back and up again.
You can teach your horse the back crunch when he has learned to lift his frontlegs on a touch of your hand or whip. You teach this exercise by asking his frontlegs to step forward while his hindlegs stay in place - and staying further and further out and behind the body. You can touch the frontleg of your horse with your whip to ask him to lift it and then gently touche the front of the leg and on the ground in front of it, to ask it to place itself back on the ground a little more forward. Then you reward him, and ask the other leg to do the same. It can take quite some time to teach to the horse, and some people therefore shorten that time by placing the foot forward with their hand or foot, but when the horse does it all by himself in his own tempo, it's safer because the back crunch is quite a heavy stretch and you don't want to force a horse into that.Edit by Romy:
The compliment (bow on one knee)
As there are wonderful instructions on how to learn doing the compliment in this thread, I will insert a picture of that too to prevent some of the confusion about the terms.
And there is another sticky that specifically deals with the compliment: Bow on one knee
New horse book: Mandala horses!
Never stop making mistakes! Natural Dressage