DA40 Instructor Pilot reported a cross threaded and improperly installed fuel manifold fitting dislodged on takeoff. This caused a complete power loss of the engine due to fuel starvation. The instructor Pilot assumed the controls and landed on the runway remaining.

Narrative

After an uneventful run-up and takeoff on runway XXL at ZZZ following outlined Diamond DA40-180 procedures found in the approved DA40 AFM; at roughly 100 AGL on climb out; established in a Vy climb the airplane experienced a sudden complete power loss from the single power plant. I immediately took controls from my student who was the pilot flying at the time. Pitched the aircraft forward to prevent the stall warning horn from activating as I saw the airspeed quickly reduce close to a speed that would induce the stall warning horn.I then told the Tower the aircraft was having engine issues and that I was going to land straight ahead. Prior to flare entry; at a stabilized airspeed of roughly 68 kts. I extended the flaps from the normal takeoff position to the landing position. I performed a normal landing with remaining runway and partially exited runway XXL at taxiway 1. Tower immediately told the aircraft on approach to the same runway behind me to go around and asked me if there was anything they could do for me. After the aircraft came to a stop I told them to contact the FBO that services our flight school to get a tug to pull us back to the ramp.After a few minutes Tower asked if we thought we could get the airplane started again. I tried several attempts; switching tanks; using and not using the boost electric pump and different throttle settings. On the initial restart attempt I immediately smelled the strong odor of 100LL avgas. Airport personnel drove up to the airplane and I helped them move the airplane off the runway. My student went back to where we initially came to a stop and noticed that there was a large amount of fuel pooled on the pavement. After a tow back to the FBO; maintenance personnel spent about an hour looking at the engine.I was told roughly 2 hours after the accident by the flight school's Chief Flight Instructor that after a swap of fuel distribution manifolds between the DA40 we were in and another flight school DA40; the maintenance technician incorrectly installed the fuel line fitting that connects the fuel line from the fuel tanks to the distribution manifold by cross threading the connection between the fuel line and the fuel line fitting. The fuel pressure produced by run-up; taxi and initial takeoff did not exceed the force required to dislodge the cross threaded connection; however; once initial takeoff power was sustained; climb power and fuel pressure were established; the connection failed. The airplane completed the corrective service and was test flown shortly after and returned to service.I regularly practice aborted takeoffs after rotation and on initial climb out with my students. Due to the training; I felt proficient exercising the actions that I performed. In the moment I was entirely focused on flying the airplane; however if I had to attempt it over again I would have tried to switch tanks possibly. Both my mixture and electric boost pump were functioning properly; as verified during the run-up and they were already confirmed several times prior to takeoff to be in the correct positions. I was unaware maintenance had been performed on the airplane; but I think from now on I will execute flights following maintenance events with extreme caution; specific relating to the initial takeoff I will use as much runway as I possibly can and will verify my engine parameters on takeoff by introducing takeoff power while holding the brakes prior to beginning my takeoff as one would normally perform for a short field takeoff.