Police have formally identified the couple who were found dead at their home in Groomsland Drive, Billingshurst on Christmas Day.
They are Jan Tshabalala, 33, and his wife Julia, 30.
A Sussex Police spokesperson said the couple have two young children, aged six and four.
The children were not believed to have been at home at the time of the incident, and are currently being cared for by relatives, police said.
A post mortem examination has confirmed that Jan Tshabalala died from hanging. The cause of death of Julia is still being investigated.
Detective chief inspector Paul Rymarz, of the Surrey and Sussex Major Crime Team, who is leading the investigation, said: “This is a tragic case and our thoughts are very much with the family at this time.
“We are not looking for anyone else in connection with the deaths but would welcome information from anyone who may have had contact with either person in the days before their deaths which may help to shed light on this case.”
DCI Rymarz added: “Because of the circumstances the couple’s families have indicated that they do not wish to speak to the media at this time and ask that their privacy is respected while they try to come to terms with their grief.”
Police are continuing to investigate the deaths. Anyone with information can email firstname.lastname@example.org or phone 101, quoting Operation Redgate.