Denis Burakov, 27, died after the ball - called a zorb - veered off course and fell into a deep gorge in the North Caucasus. His friend was badly injured.
The horrific incident at the Dombai ski resort on 3 January was filmed.
The authorities are trying to establish whether the death was caused "through providing unsafe services".
They say they want to question three owners of the zorb ride, who are reportedly on the run.
A video shot by one of Mr Burakov's friends shows how the zorb first rolls down the intended path - but then suddenly bounces off course.
A man waiting for the zorb at the bottom of the hill is then seen desperately trying to catch it, before the transparent ball falls down the nearly 1km-deep gorge below Mount Mussa-Achitara.
The ball finally stopped on a frozen lake.
Mr Burakov died of his injuries, and his friend Vladimir Shcherbov, 33, remains at hospital.
Zorbing - effectively throwing yourself down a slope in a giant ball - was invented in New Zealand in 2000 and is now a popular extreme adventure sport around the world.
It involves a giant plastic ball, which has two skins - one inside the other with air pumped in between. This acts as a shock absorber for the rider, who is in the middle of the ball.
The Russian authorities have launched an investigation after a man was killed during a ride down a ski slope inside a giant inflatable ball.