Police were called to Widmore Road, Bromley at 10.25pm on Tuesday, 14 September following reports of a stabbing.
Officers attended with London Ambulance Service paramedics and a London Air Ambulance crew.
They found a 51-year-old man with multiple stab wounds.
He was given emergency medical treatment at the scene but despite the best efforts of all involved, he was pronounced dead at 23:56hrs.
His next of kin have been informed.
At around 3.30am on 15 September, police were called to concern for the welfare of a man on the roof of an address in Burnt Ash Lane, Bromley.
Officers attended and engaged with the 29-year-old man, who later came down.
Police taser was deployed and the man was arrested on suspicion of murder. He is in custody at a south London police station.
The deceased and the man arrested are believed to be known to each other. Police are not currently searching for anyone else in connection with this incident.
A crime scene is in place and an investigation is ongoing, led by Detective Chief Inspector Chris Wood. He said: “Our thoughts are with the victim at this very difficult time. We are in the early stages of our investigation and my team are working to establish what has happened here.
“Anyone who has information that could be useful to our investigation is asked to make contact with us.”
Anyone with information is asked to call 101, giving the reference 7874/14SEP.
Information can also be provided to Crimestoppers, anonymously, on 0800 555 111.