is it possible to skate on ice without leaving a trail behind. give reasons.

No you will always leave a trail behind  You see for a skater to glide across the ice, the skater/blade must over come the static (standing) coefficient of friction. This static coefficient of friction and moving coefficient of fiction (which is always less then static friction) generate energy or heat, this heat generated instantly melts the ice as a skater travels. But because the ice and the refrigeration unit under ice, after a skater passes on, the section on which they just skated on, refreezes! 

That is what leaves a line, why a line is left, and no matter what always will!