I've noticed some startling parallels between Ancient Greek and Indian philosophy, particularly regarding reincarnation/metempsychosis, infinity, and enlightenment through purification. Did these notions develop independently? Or is there historical evidence that the two civilizations were sharing ideas?
Thomas McEvilley wrote a very interesting book examining exactly this question - "The Shape of Ancient Thought" - I recommend it.
Ultimately, of course, it is difficult to sure, the state of the evidence being what it is. There are various unverifiable claims that Pythagoras may have been influenced by Buddhism - we have almost no evidence whatsoever about him, so we can't really say.
Also a great rumor that Ammonius Sakkas - mysterious teacher of Plotinus - may have himself come from India. Again, we have no evidence at all, and at any rate this is no-longer Ancient Greece, but 'Hellenism' which is quite a lot later.
Personally I've always had the idea that Socrates and the Buddha, if they met, would get on quite well, and perhaps go on amusingly humanist adventures together.
I am writing my senior thesis now on artistic interchange between the Greco-Roman world and northern India in the Kushan dynasty -- aka Gandharan art. So I've been doing a good deal of reading on this very topic.
"Or is there historical evidence that the two civilizations were sharing ideas?"
For general information on interchange between these two civilizations in the Roman era, I would read Xinru Liu's books on the silk road. She has several, but I found "The Silk Roads: A Brief History with Documents" really useful as it contains excerpts from Pliny, Strabo, the Periplus, and other Greco-Roman authors on trade with India. Another good book on the silk road is "Rome and the Distant East: Trade Routes to the Ancient Lands of Arabia, India and China" by Raoul McLaughlin, which you can find for free online.
As for philosophy in particular, you might be interested in Adrian Kuzminski's theory that Greek Pyrrhonism was historically linked to Madhyamika thought: http://bmcr.brynmawr.edu/2009/2009-05-44.html http://enlight.lib.ntu.edu.tw/FULLTEXT/JR-PHIL/phil149570.pdf
I'm late, but there is an excellent book called 'the great transformation' by Karen Armstrong. It's about the Axial Age when Buddha, Socrates, and several others were having the same ideas on compassion, logic, etc. I read it for an anthropology of religion class.