After reaching a peak altitude of 222 km the Ariane 5 launcher with Intelsat 29e starts to lose altitude. What is the reason for this trajectory?
Altitude drops like that are common when the orbital stage has a high-efficiency, low-thrust engine.
It takes a few minutes for the upper stage to bring the craft up to orbital speed. During that period, the craft is indeed starting to fall back towards Earth. The rocket's travel over the curvature of the Earth contributes an effective altitude gain that increases as the rocket's speed increases, slowing the fall. At circular orbit speed, the fall and the curvature cancel out exactly.
The lower the acceleration of the stage that does the orbital insertion, the more dramatic the fall is.
In this case, the HM-7B engine with a thrust of 67kN is pushing the 19 ton ESC-A second stage -- mostly fuel -- plus more than 7 tons of payload. This yields roughly 0.25 g of acceleration at the start of the burn, increasing to something like 0.65 g at the end.
This is in contrast to something like Falcon 9, where the second stage has an initial acceleration around 0.75g, increasing to 3g or so as fuel runs out. That would produce a much shallower dip, if any.
We know that the tough part of getting to space isn't getting there, it's going fast enough to stay there.
One of the key reasons is that the rocket is concentrating more on gaining horizontal speed than vertical height. So the rocket will eventually start being pulled down vertically. The atmosphere is such at that altitude that the rocket won't be affected by it much, so it continues to function the same. Eventually, when orbital speeds are approached, the small vertical motion will keep the spacecraft accelerating upward, in addition to having the nearly orbital horizontal speed. And then orbit is achieved, with ever increasing vertical altitude.