Let B, Y be smooth manifolds with dim B=dim Y and let Imm( B, Y) be the space of smooth immersions of B into Y. Let M be a smooth submanifold of B and let r: Imm(B, Y)→ Imm(M, Y) be the map induced by restriction. The question of when r is a fibration is now of interest in Continuum Mechanics. Let Imm x ( M, Y) denote the immersions with normal crossings, and let S= r −1( Imm x(M, Y)) . Using the stability of Imm x ( M, Y) in C ∞( M, Y) under a group action, we show that r ̂ :S→ Imm x(M, Y) is a locally trivial fibration. The result indicates that any obstruction to r being a locally trivial fibration lies at the fibers of those immersions of M into Y which violate a transversality condition.