Lukashenko said that security measures were put in place along Belarus' border with Russia when it became apparent, after the attack on the Crocus City Hall on Friday, that the perpetrators had driven a car into the Russian region of Bryansk, which borders Belarus and Ukraine.
The authoritarian long-term ruler of the ex-Soviet republic, which is allied with Russia, said that the attackers "were therefore unable to enter Belarus. They saw that. That's why they turned around and drove towards the Ukrainian-Russian border".
Russia on Tuesday remanded in custody a Kyrgyzstan-born man suspected of involvement in the mass shooting and sent investigators to Tajikistan to question the families of four men charged with carrying out the attack.
Eight suspects have been remanded in pre-trial detention since gunmen sprayed concertgoers with bullets in the deadliest attack in Russia in two decades, days after President Vladimir Putin celebrated an election that handed him a fifth term.
The Islamic State group has claimed responsibility and released footage from the attack.
The United States and France say intelligence suggests the group was indeed behind the attack, in which 139 people were killed and 182 wounded.
Putin said on Monday the attack had been carried out by Islamic militants but also suggested that Ukraine, which is at war with Russia, may have played a role.
Ukraine has denied any role.
The militants have not identified any of the attackers.
Russia has said the four suspected gunmen have confessed but some showed signs of injuries when they appeared in court, raising concern they had been tortured.
Russia's commissioner for human rights said detention of suspects should be carried out in accordance with the law, TASS news agency reported, after videos were published showing the interrogation of the suspects.
Human rights commissioner Tatyana Moskalkova says torture must not be used on detainees. (EPA PHOTO)
One had part of his ear cut off during questioning.
"It is absolutely unacceptable to use torture on detainees and defendants," the commissioner, Tatyana Moskalkova, was quoted as saying by TASS.
Russian authorities have said they are investigating.
Three Tajik sources told Reuters on Tuesday that Russian investigators were in Tajikistan questioning the families of the four suspected gunmen, saying their relatives had been brought to the capital Dushanbe from their home towns.
On Tuesday, Putin said he hoped prosecutors would do everything to ensure the attackers would be justly punished.
Alisher Kasimov is accused of providing accommodation to four shooting suspects. (EPA PHOTO)
Kyrgyzstan-born Alisher Kasimov, remanded in custody on Tuesday, was led into the court room bent double - like the other suspects - before his handcuffs were removed.
He showed no visible signs of injury.
He is accused of providing accommodation to the four Tajik men accused of carrying out the attack.
The Tajik sources said Tajik President Emomali Rakhmon was personally overseeing the Tajik part of the investigation into the Moscow attack, which he said was a "terrible and shameful event".
Russian investigators say that after firing from Kalashnikov AK-47 weapons, the attackers set fire to the building with petrol before leaving, hitting a family with two young children as they sped out of the car park.
Earlier this month, Rakhmon said his government was alarmed by the activity of radical Islamist preachers who were "brainwashing" Tajik youths, making them susceptible to manipulation by foreign groups and intelligence agencies.
with DPA
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