Looking at the history of Sri Lanka, India has greatly influenced the Ancient Sri Lanka. Sri Lanka's politics, culture, society, etc. have been heavily influenced by India. After the 16th century, the European influence has been applied to the cultural sphere due to the western colonization.
To understand you have to take into account the religions, the ethnic origin, and the commercial links. Buddhism is dominant in its Theravada form and Sri Lanka spread it to Southeast Asia from Myanmar to Vietnam. The Tamils are Hinduists but they play a very important technical role since it is a "caste" of theirs that devised and built the essential water managing system that was fully active in the fourth century with Sigiriya's water gardens at the foot of the rock. The water system has still not been reactivated though it is upgraded for the cultivation of rice and agriculture in general, but not with Indian technicians. But the Muslims were essential for maritime commerce, but not the Arabic side, in fact, the Iranian side. Quite a few Arabian Nights tales include Sri Lanka in one way or another and I saw how Sinbad the Sailor was popular in Sri Lanka in 2005. The three religions, and three languages are present in Zheng He's Plaque in Galle: Buddhism, Hinduism, Islam, on one hand, and Chinese (though it could have been Pali or Sinhala), Tamil and Iranian, on the other hand. To reduce it to only one influence, India, is restrictive, just the same, with Europeans. First the Portuguese for three centuries with a great number of family names that are Portuguese and the presence of the Catholic faith (note the Anglican religion will never be really present when the English took over). Then the English who increased the presence of Tamils as plantation workers with the intention to divide the population. But Sri Lanka seems to have been spared from the slave trade from eastern Africa to the Mediterranean (later Muslim) world and then to India. We have to keep in mind that Buddhism is hostile to castes and to slavery.