Using whatever POD you think best accomplishes it, imagine a world where the Carolingian Empire does not collapse during the 9th century, but instead survives as a cohesive and powerful entity for at least several more centuries (in such a scenario it'd probably be colloquially referred to as the Roman Empire in the West and the Frankish Empire in Constantinople, but that's neither here nor there). We're talking about an empire that will rule over at least the lands depicted below (with the possibility of expansion, especially in the east) continuously for centuries.
Such a scenario could be accomplished by something as simple as different inheritance laws or certain individuals dying in childhood, or something more creative. Point is, the Western Roman Empire is revived and rules from Aachen.
What would all of this mean for the development of the Normans? IOTL Norse Vikings were very active in France, eventually establishing such an important presence that West Francia, one of the Carolingian successor states, cut a deal with Viking leader Rollo to allow his people to settle in what would become Normandy. Centuries of intermarriage and acculturation would produce the Norman peoples, who played a major role in European history.
Would a surviving and powerful Carolingian Empire be able to successfully rebuf any significant settlement by Vikings in its territory? Or would the demographic situation in that part of France still lead Aachen to a similar deal with some group of Vikings or another? If so, what effect would a surviving Carolingian Empire have upon the development of the Normans or their equivalents here? Would the continued power of the Franks largely preclude the Normans from becoming as successful as they were? Or would the Normans or their equivalents be liable to be an important force in the surviving Carolingian Empire?