Mr Mangione is now expected to be presented with the option of waiving his extradition to the state of New York, or contesting it.
Different US states have different laws and judicial systems so there is a process involved in the transfer of fugitives, which can take days or weeks.
Mr Mangione's family said they were "shocked and devastated" by his arrest, and offered their prayers to the family of Mr Thompson.
His paternal grandparents, Nicholas and Mary Mangione, were real estate developers who purchased the Turf Valley Country Club in 1978 and Hayfields Country Club in Hunt Valley in 1986.
One of his cousins is a Republican Maryland state legislator.
As a teenager, Mr Mangione attended a private all-boys school in Maryland, where he was class valedictorian, a title usually awarded to students with the best grades.
He graduated from the University of Pennsylvania, an Ivy League college.
His LinkedIn account says he worked as a data engineer in California. TrueCar, a website for car buyers, confirmed that he had been employed there but left in 2023.
Mr Mangione spent time in a co-living surfing community in Hawaii called Surfbreak.
Sarah Nehemiah, who knew him then, told CBS News he left due to his back injury which worsened when he surfed.
Several posts to an account on X, formerly Twitter, that appeared to belong to Mr Mangione suggested that friends had been trying to reach him, with one person posting in October that "nobody has heard from you in months".