ormal ▁definition ▁ ▁In ▁the ▁following ▁we ▁assume ▁all ▁manif olds ▁are ▁differenti able ▁manif olds ▁of ▁class ▁Cr ▁for ▁a ▁fixed ▁r ▁ ≥ ▁ 1 , ▁and ▁all ▁morph isms ▁are ▁differenti able ▁of ▁class ▁Cr . ▁ ▁Im mers ed ▁sub man if olds ▁ ▁An ▁imm ers ed ▁sub man if old ▁of ▁a ▁manifold ▁M ▁is ▁the ▁image ▁S ▁of ▁an ▁imm ersion ▁map ▁f : ▁N ▁→ ▁M ; ▁in ▁general ▁this ▁image ▁will ▁not ▁be ▁a ▁sub man if old ▁as ▁a ▁subset , ▁and ▁an ▁imm ersion ▁map ▁need ▁not ▁even ▁be ▁inject ive ▁( one - to - one ) ▁– ▁it ▁can ▁have ▁self - inter sections . ▁ ▁More ▁narrow ly , ▁one ▁can ▁require ▁that ▁the ▁map ▁f : ▁N ▁→ ▁M ▁be ▁an ▁injection ▁( one - to - one ), ▁in ▁which ▁we ▁call ▁it ▁an ▁inject ive ▁imm ersion , ▁and ▁define ▁an ▁imm ers ed ▁sub man if old ▁to ▁be ▁the ▁image ▁subset ▁S ▁together ▁with ▁a ▁topology ▁and ▁differential ▁structure ▁such ▁that ▁S ▁is ▁a ▁manifold ▁and ▁the ▁inclusion ▁f ▁is ▁a ▁dif fe omorphism : ▁this ▁is ▁just ▁the ▁topology ▁on ▁N , ▁which ▁in ▁general ▁will ▁not ▁agree ▁with ▁the ▁subset ▁topology : ▁in ▁general ▁the ▁subset ▁S ▁is ▁not ▁a ▁sub man if old ▁of ▁M , ▁in ▁the ▁subset ▁topology . ▁ ▁Given ▁any ▁inject ive ▁imm ersion ▁f ▁: ▁N ▁→ ▁M ▁the ▁image ▁of ▁N ▁in ▁M ▁can ▁be