India is home to millions of Tamils in Tamil Nadu, separated by a narrow strip of sea from Sri Lanka.
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Prime Minister Manmohan Singh said Rajapakse's win in Tuesday's election reflected "the trust the people of Sri Lanka have reposed" in him.
"I am confident that, under your leadership, Sri Lanka will find lasting peace, where all communities can live with dignity and in harmony," Singh said in a congratulatory letter.
The presidential poll was the first since the defeat of the separatist Tamil rebels last May that ended a decades-long civil war.
The election had been a straight race between Rajapakse and former army chief Sarath Fonseka -- the two men credited with scripting the decisive win over the rebel Liberation Tigers of Tamil Eelam (LTTE), who fought for 37 years for a Tamil homeland.
In a separate statement, the Indian foreign ministry said: "The peaceful conduct of elections augurs well for the future of Sri Lanka and we hope that the post-election scenario would be free of vi