Sri Lanka's foreign ministry said it had "lodged strong protests with the Indian Government" through its diplomatic missions in New Delhi and Chennai.
India's High Commission (embassy) in Colombo said the central government in consultation with state governments was taking action to ensure the safety of Sri Lankan tourists.
"The state government of Tamil Nadu has taken immediate action to identify and prosecute the miscreants, in respect of the recent incidents," the High commission said.
"These isolated incidents do not reflect the strong people-to-people bonds that have been an integral part of the close historical, cultural and civilisational ties between India and Sri Lanka."
Last year 250,000 Sri Lankans had visited India.
Sri Lanka's foreign ministry asked travellers to "exercise caution when undertaking visits to the State of Tamil Nadu in India," and to contact the Sri Lanka mission in Chennai if visits were still made "due to compelling reasons.
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The foreign minis