The rise of the Bactrian and Indo-Greek kingdoms following the conquest by Alexander the Great brought about significant cultural exchange between India and the Hellenistic Kingdoms.
We know some Greeks converted to Buddhism, and Buddhist monks were known to have travelled to Alexandria in Egypt, and its argued introduced a philosophical and religious shift. Why was Buddhism so influential and not Hinduism or Jainism?
EDIT: I have maybe been premature in asking this question. There are some problems with the assumptions in my question. Firstly, it’s conceivable that there was earlier contact between the Greeks and Indians in the Persian empire. Secondly, after spending the morning reading more critically, I now understand that perhaps the only indication we have of Buddhist monks in Alexandria are the pillars of Ashoka in India and even then it could be argued he meant ambassadors rather than monks. The Greeks who are known to have converted were from the Bactrian and Indo-Greek kingdoms, or from Greco-Buddhist art, in other words only those right on the doorstep of India. Finally, I’m very dubious that Buddhism had much of an effect on philosophy or religion in the Hellenistic World or later, rather than similar ideas appearing in different places within 200 or so years of each other, although I am open to the idea that Pyrrho may have been intrigued by ideas he picked up while in the court of Alexander the Great in India.
Finally, I now appreciate that the Sauryan empire conquered and dominated the Bactrian kingdom (the Indo-Greek kingdoms only emerging again as the empire began to collapse) so it absolutely made sense for the Bactrian Greek elite to align themselves with the conquering court, and the adoption of Buddhism would be part of this, especially after Ashoka adopted the religion.
So what I would ask is what do we know of the Indian gymnosophists mentioned by ancient writers and did they represent Buddhism, Jainism or Hinduism (or indeed some other Indian religion)?
Why was Buddhism so influential and not Hinduism or Jainism?
Prior to Buddhism spreading outside India, Hinduism in the form of Brahmanism actually spread throughout a lot of mainland and insular Southeast Asia. Angkor Wat was probably originally constructed as a Hindu temple, and there are still vestiges of Brahmanism and Hindu practice in other parts of Southeast Asia- shrines to Brahma in Thailand, Hindu culture in Bali, etc. Pre-Buddhist contact with the West was probably minimal during the interval between the Indus Valley Civilization and the Second Urbanization, because the trade routes involved were longer and more difficult, and as the center of urbanization moved into eastern India and the Ganges basin it was natural for trade to focus on SE Asia rather than Persia and the West.
The question of why Buddhism superseded Brahmanism outside of India is complicated, and I don't know that anyone claims to have a comprehensive answer. One is that Buddhism didn't rely on a hereditary priesthood- you could create monks and nuns from the local population just by importing a small quorum of monks, whereas brahmins had to be brought in with their families and limited their priesthood to people born into brahmanic lineages. Buddhism also seems to have been popular with traders and merchants who traveled between India and SE Asia. Finally, while there were pilgrimage sights associated with the Buddha, sacred geography plays a much smaller role in Buddhism- sacred rivers and mountains (or their architectural substitutes) were not regarded as central to the establishment of the religion, or as necessary parts of ritual life.
The usual answer given for why Jainism didn't spread in the way that Buddhism did is that Jain monks have much stricter beliefs regarding travel- they generally do not undertake long journeys because it is believed to cause the death of small living creatures by being crushed by vehicles. Jain regulations on eating may have made it more difficult to travel long distances (meat and fish being something that is easy to salt or smoke for a long overland or sea voyage). Despite this, Jain monks are one of the likely candidates for the 'gymnosophists' of Alexandria because one sect of Jain monks traditionally goes about naked rather than wearing robes. Even if this was the case, the relative austerity of Jain practice may have made it more difficult for them to sustain communities outside India.
Identities of possible Indian groups in the West in antiquity are difficult to ascertain. One example is the issue of the Theraputae of Alexandria, a group we have information about from an Alexandrian Jewish philosopher, but whose exact identity is unknown. Some Christians see them as an early Christian group, but the idea that they were Buddhist monks (and that their name was somehow related to 'Theravada') has been floated by other historians. There were also other sramana movements (like the Ajivikas) who were present in India and could have spread beyond it, but who left no independent records of their own and are known only through potentially hostile descriptions from Buddhist or Jain sources. Ancient sources usually record little of the philosophy of these groups that is very specific, and what is there can't necessarily be accepted uncritically as accurate- writers may have been relying on summaries they got from others, and there are issues of language and translation that are still thorny in the modern world. Usually, descriptions focus on ascetic lifestyles, which make it difficult to identify any particular group, since Buddhists, Jains, and Brahmin ascetics all seem to have adopted similar principles that were drawn from a pre-existing tradition of asceticism in India.
Christopher Beckwith's book Greek Buddha raises the possibility that Buddhism may have influenced Greek philosophy via the transmission of fundamental Buddhist teachings West via a translator affiliated with Alexander who was able to learn local languages and customs and return them to Greece. The difficulty here is that we don't have the original writings of this person, Pyrrho of Elis, but rather a third-hand description of his teachings according to a leading disciple, as preserved by Aristocles. Beckwith makes an interesting case that some of the principles of Greek skepticism might ultimately have been drawn from early Buddhist views, but I don't know how widely his work has been accepted- he also claims on etymological grounds that Lao Tzu was actually the Buddha, and that the Tao Te Ching is an early Chinese attempt at representing Buddhist teachings that were already circulating outside India, which is definitely an unorthodox take.