Dr Kalburgi, a former vice chancellor of Hampi University, was shot dead on his doorstep in Dharwad town by men aged between 20 and 30 years, who came in a bike.
The family of the 77-year-old rationalist has alleged that he was killed for his beliefs.
The professor had received many threats to his life because of his comments on religious issues, and had also faced protests by pro-Hindu activists.
A leader of the Vishwa Hindu Parishad (VHP) today asserted that no rightwing group had threatened Dr Kalburgi.
"We condemn this killing, I guarantee you that no rightwing organization is involved in the killing," said Ramesh Kulkarni, a VHP state secretary, adding, "We train to protest, not to kill."
Because of the threats, Dr Kalburgi was given an armed escort but the cover was withdrawn a year and a half ago. "My father didn't want to stop living for fear of death," his daughter Rupadhiri said yesterday.
The professor had provoked the anger of some groups by speaking in support of writer UR Ananthamurthy when he shared that as a child he had urinated on stone figurines. Dr Kalburgi's comment that the Lingayat community that he belonged to, is not Hindu, was also condemned by his critics, who called his writings and speeches "anti-Hindu."
"There were threats to Kalburgi from us but those were threats of boycotting him, there were no threats to his life from us," said Mr Kulkarni.
The police have filed a case against an activist of Bajrang Dal, an affiliate of the VHP, for a threatening tweet after the killing. The Bajrang Dal and the VHP are both a part of a cluster called the "Sangh Parivar", led by the BJP's ideological mentor Rashtriya Swayamsevak Sangh.
The state's Congress government has said it will ask the CBI to investigate the murder that has led to protests in Dharwad and other parts of Karnataka.
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