An immersion is a smooth map 
between smooth manifolds
whose tangent map 
is injective
at every point 
. If 
and 
, this requires 
. In suitable coordinate
charts around 
and 
, the map
has the local form 
(Balzin 2021, pp. 38-41). The coordinate charts
are on the manifolds, rather than on their tangent
spaces.
An immersion need not be one-to-one or an embedding. For example, the circle parametrized by 
modulo 
has an immersion into 
given by 
. Its derivative 
is never zero, but 
, producing a self-intersection.
An isometric immersion preserves the Riemannian metric on tangent vectors.
