I wouldn't say there were deep ties, but there certainly were some. Only older people remember those ties now. You will find comments by such people on Youtube clips of old Hindustani songs. Not just Russia, they come from other ex-Soviet states as well. I've talked to some younger Russians, and they can sometimes recognize these tunes, but have no idea about their origin.
[1955] https://www.youtube.com/watch?v=rW72kDrJvwA
[1955] https://www.youtube.com/watch?v=5wjGc1zGWBc
[1955] https://www.youtube.com/watch?v=PXxR1Fj-BQ0
[1955] https://www.youtube.com/watch?v=oXLzfldeDcM
[1956] https://www.youtube.com/watch?v=vwZVFw7bhos