To expose themselves, their underside in general, their genitalia in particular.
While “submission” may at first appear to be fundamentally different from “dominance,” in reality they are opposite and yet equal expressions of the same urge to make contact (indirectly, i.e. via sexual contact) with something they are attracted to but, have associated a strong feeling of resistance toward. A state of sexual arousal is the common denominator in both reflexes tracing back to the same physical memory of being stimulated by their mother as infant puppies when they were unable to eliminate on their own. We could say they are presenting themselves to be cleaned.
But on a deeper level what we call submissiveness is actually dogs acting “prey-like.”
Why would a submissive dog act prey-like: what’s the advantage to this behavior? Because in the natural scheme of things, the prey-controls-the-predator by which I mean that the predator can’t act like a predator unless the prey acts like the prey.
When a dog rolls over, tail wagging intensely side-to-side, squirting urine, lip licking, it is “vibrating” like prey and being emotionally attractive gives it leverage, if that is, it can feel how to exploit being the object-of-attraction. (Watch a cat train a dog if one wants to see a master in action.) Once a dog feels its leverage, it then quickly learns to press in and exercise its “control” over its so-called superior. So while we tend to think of the dominant animal as in control over the so-called submissive one, we would be well advised to look again. Who is really in control of whom?
By going belly up and vibrating intensely, and then pressing in on the dominant individual, the “prey-like” individual is taking over the dominant dog’s nervous system. This is exactly how infant puppies enervate and “control” adults. The mother is emotionally attracted to the pup given that it is the pure embodiment of prey energy, but then the infant pup acts like a predator, i.e. it goes toward the mother and squeals when disturbed (prey animals typically die in silence which is why the American Indian revered the deer, as opposed to the bear or mountain lion, as the model in stoicism and courage that warriors should emulate) and so the mother is induced to tend to it in order to keep it calm.
On the other hand if the pup doesn’t generate a predatory nature to inhibit the mother’s urge to make prey on it, it is therefore sickly and the mother will eat it. But in the case of healthy pups, by ingesting the placenta and umbilical cord, and then thereafter the urine and feces, the mother’s attraction to the pup as prey is satisfied and so their relationship will continue to evolve into a deep emotional bond as the pup matures.
And so it goes full circle. As adults the softer natured dogs when they become the object-of-attention in the face of pressure; roll over to expose themselves in order to gain “control.”