Definition
Immersion and Embedding
A smooth map $f : M \to N$ (Definition~Smooth Map and Diffeomorphism) is an immersion if the differential $df_p : T_pM \to T_{f(p)}N$ (Definition~Tangent Space) is injective for all $p$.
It is an embedding if it is additionally a homeomorphism (Definition~Continuous Map and Homeomorphism) onto its image.
An embedded submanifold is the image of an embedding; an immersed submanifold is the image of an injective immersion (which need not be a homeomorphism).
Used in
Dependency Graph
flowchart LR
classDef current fill:#6366f1,color:#fff,stroke:#4f46e5
n03a51003["Smooth Map and Diffeomorphism"]
nd0ae9dbd["Tangent Space"]
na9fa06fb["Continuous Map and Homeomorphism"]
ne5d64c7b["Immersion and Embedding"]:::current
n9bfcb403["Remark"]
n03a51003 --> ne5d64c7b
nd0ae9dbd --> ne5d64c7b
na9fa06fb --> ne5d64c7b
ne5d64c7b --> n9bfcb403
click n03a51003 "../objects/03a51003.html" "_self"
click nd0ae9dbd "../objects/d0ae9dbd.html" "_self"
click na9fa06fb "../objects/a9fa06fb.html" "_self"
click n9bfcb403 "../objects/9bfcb403.html" "_self"