Nostrum I believe means ours. Nostrum mare, our sea. Yes, mediterranean, between the lands.
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Well, I am not a specialist in Indian architecture, but the arhchitectural tradition of the medieval Russia obviously comes from Byzantium cathedrals. There are traditional golden domes in both cultures- sometimes it has three- for the Holy Trinity, sometimes five- for Christ and 4 authors of recognized Scriptures- i.e there can be variations. There were practically no links between Russia and China/India until modern times. I am just an amateur on the topic, so can't contribute much...
Wasn't St Petersburg a New Town, established from scratch by Peter the Great as his 'Window on the West'? He toured Europe, I believe, and took back a very definite idea of what he wanted his Brave New City to look like - as much like Versailles as possible - as well as how he wanted to haul the court into modern Europe, willy nilly.
Perhaps the strongest link between Russia, China, India, Iran and Turkey was through Tamerlane the Great who held sway over all these territories in the 14th century. His center of power was at Samerkand which is right in the middle of this huge area.
For even earlier links there is the famed Silk Road connecting China with Central Asia, eastern europe and even Africa. There are accounts of Greek explorer/traders reaching China in around 200 BC.