Daniel Young, 30, was attacked in Kendor Gardens near Morden Underground station shortly after 08:00 GMT on Tuesday.
A post-mortem examination found he died from a stab wound.
A 35-year-old man has been arrested on suspicion of murder and is currently in custody.
On the day of the attack a 26-year-old man was arrested on suspicion of murder and wasting police time. He has been bailed.
Mr Young, originally from Paignton in Devon, was attacked as he made his way to the London Campus of Coventry University for the second day of a new job.
He was attacked in a park used by commuters as a short cut to Morden Tube station and Morden Road tram stop.
It happened less than 40 minutes after a 23-year-old man was stabbed in the buttocks in nearby Lewis Road.
Scotland Yard said it was investigating whether the two attacks were linked and trying to establish the motive and whether the attacker was known to Mr Young.
Meanwhile, an online fundraising page apparently set up in memory of Mr Young has been removed after his family reportedly said they had no knowledge of it.
Police said they understood that the donations made through the Gofundme page would be refunded. The Met's Cyber Crime Unit is investigating.
On Facebook, Mr Young's brother Simon wrote: "My Brother. My friend. I'm still wrapping my head round this, still hoping/wishing it's a sick joke.
"I don't know what to feel, I'm frustrated, angry, upset, confused. Just so proud of everything you achieved in your life, places you travelled, friends you made...you are inspirational big Bro."
Det Supt Paul Monk said: "I understand the concerns of the public that the attack on Daniel Young was unprovoked and potentially committed by a stranger in a random attack.
"We too share those concerns and have allocated significant resources to find the person who did this."

A judge is deciding if there is enough evidence for the case against Seamus Daly, 45, from Jonesborough, County Armagh, to go to a crown court trial.
Mr Daly, originally from the Republic of Ireland, was arrested in April 2014.
The Omagh bomb was the biggest single atrocity in the history of the Troubles in Northern Ireland.
It happened just four months after the Good Friday Agreement was signed.
The death toll included nine children and three generations of one family, but no-one has been convicted in a criminal court of carrying out the attack.
The hearing at Omagh Magistrates' Court is expected to last for two days.
Along with the murder charges, Mr Daly faces charges of causing the explosion and possessing the bomb, and two charges relating to another dissident republican bomb plot in Lisburn, County Antrim, in April 1998.
Previously, all of Mr Daly's magistrates' court hearings in Omagh have been via videolink from prison.
He was brought to court in a prison van on Thursday and was led inside handcuffed to a security officer.
Some relatives of those who were killed in the attack were in the court.
Mr Daly sat a few feet from the victims' relatives who have followed every stage of the long legal process.
After his arrest Mr Daly gave police a statement denying any involvement in the Omagh bombing or Lisburn plot.
His lawyers have argued that the case against him is weak and much of the evidence has been discredited.
A second man has been arrested in connection with the murder of a university lecturer killed near a London Tube station.
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A man accused of murdering 29 people in the Real IRA bomb attack in Omagh in 1998 has appeared in court in the County Tyrone town for the first time.