Does it make sense to do an emergency landing without nose gear in order to reduce stopping distance? Would a pilot decide to land without the nose gear down to increase friction if for example all the engines have suffered a flame-out and there is only a very short runway available ?
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That was the case on July 23, 1983 (but not quite).

It was luck the manual gravity-extension didn't lock the nose gear in position. That has been attributed to the shortening of the landing roll/slide.


  Two factors helped avert a potential disaster: the failure of the front landing gear to lock into position during the gravity drop, and the presence of a guardrail that had been installed along the centre of the decommissioned runway to facilitate its use as a racetrack.


However, no commercial aircraft is certified for it, because it can do more harm than good. And depending on the aircraft type, separate controls for each landing gear in manual-extension mode may not exist.

A hard landing without the nose-landing gear may rupture the fuselage, center fuel tank, or hydraulic lines. And with the added sparks from the sliding, that's a recipe for disaster.

Instead, a system such as the engineered materials arrestor system (EMAS) may help:

(Source) EMAS.