A non-injective immersion:
R →
R 2 ,
t ↦ (
t 2 - 1,
t (
t 2 - 1))
In differential topology , immersion is understood to be a smooth mapping between manifolds and when the push forward of this mapping is injective at every point . If there is also a topological embedding , one speaks of a (smooth) embedding . In this case the image of the map is too diffeomorphic a submanifold of
F.
:
M.
→
N
{\ displaystyle F \ colon M \ rightarrow N}
M.
{\ displaystyle M}
N
{\ displaystyle N}
F.
∗
p
:
T
p
M.
→
T
F.
(
p
)
N
{\ displaystyle F _ {\ ast p} \ colon T_ {p} M \ to T_ {F (p)} N}
p
∈
M.
{\ displaystyle p \ in M}
F.
{\ displaystyle F}
M.
{\ displaystyle M}
N
.
{\ displaystyle N.}
The properties of the image in the general case are described in the entry Immersed Manifold .
Immersion in Euclidean space
If there is the special case of a mapping between Euclidean spaces, then it represents nothing other than the total derivative or the Jacobi matrix , whereby the Euclidean space is identified in a natural way with its tangential space and a linear mapping with a matrix.
F.
:
R.
m
→
R.
n
{\ displaystyle F: \ mathbb {R} ^ {m} \ rightarrow \ mathbb {R} ^ {n}}
F.
∗
:
T
p
R.
m
→
T
F.
(
p
)
R.
n
{\ displaystyle F _ {\ ast}: T_ {p} \ mathbb {R} ^ {m} \ rightarrow T_ {F (p)} \ mathbb {R} ^ {n}}
D.
F.
(
p
)
:
R.
m
→
R.
n
{\ displaystyle DF (p) \ colon \ mathbb {R} ^ {m} \ rightarrow \ mathbb {R} ^ {n}}
Immersion in manifolds
In general, a differentiable mapping is an immersion if and only if the rank of the linear mapping is equal to the dimension of the manifold for all of them , that is, it applies
F.
:
M.
→
N
{\ displaystyle F: M \ rightarrow N}
p
∈
M.
{\ displaystyle p \ in M}
F.
∗
{\ displaystyle F _ {\ ast}}
M.
{\ displaystyle M}
rank
F.
p
=
dim
(
image
(
F.
∗
p
)
)
=
dim
M.
.
{\ displaystyle \ operatorname {rank} F_ {p} = \ dim (\ operatorname {image} (F _ {\ ast p})) = \ dim M.}
Regular homotopy
Two immersions are regularly called homotopic if there is a homotopy with such that the mapping
for each
F.
0
,
F.
1
:
M.
→
N
{\ displaystyle F_ {0}, F_ {1} \ colon M \ to N}
F.
:
M.
×
[
0
,
1
]
→
N
{\ displaystyle F \ colon M \ times \ left [0,1 \ right] \ to N}
F.
(
m
,
0
)
=
F.
0
(
m
)
,
F.
(
m
,
1
)
=
F.
1
(
m
)
∀
m
∈
M.
{\ displaystyle F (m, 0) = F_ {0} (m), F (m, 1) = F_ {1} (m) \ forall m \ in M}
t
∈
[
0
,
1
]
{\ displaystyle t \ in \ left [0,1 \ right]}
F.
t
:
M.
→
N
{\ displaystyle F_ {t} \ colon M \ to N}
F.
t
(
m
)
=
F.
(
m
,
t
)
{\ displaystyle F_ {t} (m) = F (m, t)}
is an immersion again.
The Hirsch-Smale theory deals with the regular homotopy classes of immersions .
See also
literature
John M. Lee: Introduction to Smooth Manifolds (= Graduate Texts in Mathematics 218). Springer-Verlag, New York NY et al. 2003, ISBN 0-387-95448-1 .
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