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A continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i0502501.png" /> of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i0502502.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i0502503.png" /> into an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i0502504.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i0502505.png" /> such that for each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i0502506.png" /> there exists a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i0502507.png" /> in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i0502508.png" /> is an imbedding, i.e. a [[Homeomorphism|homeomorphism]] onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i0502509.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025010.png" /> is a homeomorphism into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025011.png" />, then it is called an imbedding of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025012.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025013.png" />. The immersion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025014.png" /> is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025016.png" />-immersion if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025018.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025021.png" />-(smooth) manifolds (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025024.png" />) and if the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025025.png" /> on the corresponding charts is given by functions
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025026.png" /></td> </tr></table>
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that belong to the smoothness class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025027.png" />, while the rank of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025028.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025029.png" /> at each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025030.png" /> (a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025031.png" />-(smooth) manifold is a manifold provided with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025032.png" />-structure, where the pseudo-group consists of mappings that are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025033.png" />-times differentiable and whose derivatives satisfy the Hölder condition of index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025034.png" />).
+
A continuous mapping  $  F: M  ^ {m} \rightarrow N  ^ {n} $
 +
of an  $  m $-
 +
dimensional manifold  $  M  ^ {m} $
 +
into an  $  n $-
 +
dimensional manifold  $  N  ^ {n} $
 +
such that for each point  $  x \in M  ^ {m} $
 +
there exists a neighbourhood  $  U _ {x} $
 +
in which  $  F $
 +
is an imbedding, i.e. a [[Homeomorphism|homeomorphism]] onto  $  F ( U _ {x} ) \subset  N  ^ {n} $.  
 +
In particular, if  $  F $
 +
is a homeomorphism into  $  F ( M  ^ {m} ) $,
 +
then it is called an imbedding of  $  M  ^ {m} $
 +
in  $  N  ^ {n} $.
 +
The immersion  $  F $
 +
is called a $  C ^ {l, \alpha } $-
 +
immersion if  $  M  ^ {m} $
 +
and  $  N  ^ {n} $
 +
are  $  C ^ {l, \alpha } $-(
 +
smooth) manifolds ( $  l \geq  1 $,
 +
$  0 \leq  \alpha < 1 $,
 +
$  m \leq  n $)
 +
and if the mapping  $  F $
 +
on the corresponding charts is given by functions
  
The concepts of a surface and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025035.png" />-(smooth) surface are closely related to the concepts of an immersion and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025036.png" />-(smooth) immersion. Two immersions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025038.png" /> between manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025040.png" /> are called equivalent if there is a homeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025041.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025042.png" />.
+
$$
 +
x  ^ {i= f ^ { i } ( u  ^ {1} \dots u  ^ {m} ),
 +
i = 1 \dots n,
 +
$$
  
An immersed manifold is a pair consisting of a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025043.png" /> and an immersion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025044.png" /> of it. A surface of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025046.png" /> in a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025047.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025048.png" /> is a class of equivalent immersions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025049.png" />; each immersion of this class is called a parametrization of the surface. A surface is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025052.png" />-smooth if one can introduce <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025053.png" />-structures in the manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025055.png" /> and if among the parametrizations of the surface one can find a parametrization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025056.png" /> which in these structures is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025057.png" />-immersion.
+
that belong to the smoothness class  $  C ^ {l, \alpha } $,
 +
while the rank of the matrix  $  \| df ^ { i } / du  ^ {j} \| $
 +
is equal to  $  m $
 +
at each point  $  x \in M  ^ {m} $(
 +
a  $  C ^ {l, \alpha } $-(
 +
smooth) manifold is a manifold provided with a  $  \Gamma $-
 +
structure, where the pseudo-group consists of mappings that are  $  l $-
 +
times differentiable and whose derivatives satisfy the Hölder condition of index  $  \alpha $).
 +
 
 +
The concepts of a surface and a  $  C ^ {l, \alpha } $-(
 +
smooth) surface are closely related to the concepts of an immersion and a  $  C ^ {l, \alpha } $-(
 +
smooth) immersion. Two immersions  $  F $
 +
and  $  G $
 +
between manifolds  $  M $
 +
and  $  N $
 +
are called equivalent if there is a homeomorphism  $  \Phi :  M \rightarrow M $
 +
such that  $  F = G \Phi $.
 +
 
 +
An immersed manifold is a pair consisting of a manifold $  M $
 +
and an immersion $  F $
 +
of it. A surface of dimension $  m $
 +
in a manifold $  N  ^ {n} $
 +
of dimension $  n $
 +
is a class of equivalent immersions $  F: M  ^ {m} \rightarrow N  ^ {n} $;  
 +
each immersion of this class is called a parametrization of the surface. A surface is called $  C ^ {l, \alpha } $-
 +
smooth if one can introduce $  C ^ {l, \alpha } $-
 +
structures in the manifolds $  M $
 +
and $  N $
 +
and if among the parametrizations of the surface one can find a parametrization $  F $
 +
which in these structures is a $  C ^ {l, \alpha } $-
 +
immersion.
  
 
The theory of immersed manifolds usually deals with properties that are invariant under the above concept of equivalence, and in essence coincides with the theory of surfaces, particularly when one considers topics related to the geometry of immersions.
 
The theory of immersed manifolds usually deals with properties that are invariant under the above concept of equivalence, and in essence coincides with the theory of surfaces, particularly when one considers topics related to the geometry of immersions.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025058.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025059.png" />-manifold, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025061.png" />. Any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025062.png" /> allows for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025063.png" /> an imbedding into the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025064.png" /> and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025065.png" />-immersion into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025066.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025067.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025068.png" /> is positive and not a power of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025069.png" />, then any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025070.png" /> allows a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025071.png" />-imbedding into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025072.png" />, whereas for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025073.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025074.png" /> there exist closed smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025075.png" />-dimensional manifolds not allowing even a topological imbedding into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025076.png" /> (such as, for example, a projective space). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025077.png" /> does not have compact components, it allows a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025078.png" />-imbedding into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025079.png" />.
+
Let $  M  ^ {m} $
 +
be a $  C ^ {l, \alpha } $-
 +
manifold, $  l \geq  1 $,
 +
0 \leq  \alpha < 1 $.  
 +
Any $  M  ^ {m} $
 +
allows for $  m \geq  1 $
 +
an imbedding into the Euclidean space $  \mathbf R  ^ {2m} $
 +
and a $  C ^ {l, \alpha } $-
 +
immersion into $  \mathbf R  ^ {2m-} 1 $
 +
for $  m \geq  2 $.  
 +
If $  m $
 +
is positive and not a power of $  2 $,  
 +
then any $  M  ^ {m} $
 +
allows a $  C ^ {l, \alpha } $-
 +
imbedding into $  \mathbf R  ^ {2m-} 1 $,  
 +
whereas for any $  m = 2  ^ {s} $
 +
with $  s \geq  0 $
 +
there exist closed smooth $  m $-
 +
dimensional manifolds not allowing even a topological imbedding into $  \mathbf R  ^ {2m-} 1 $(
 +
such as, for example, a projective space). If $  M  ^ {m} $
 +
does not have compact components, it allows a $  C ^ {l, \alpha } $-
 +
imbedding into $  \mathbf R  ^ {2m-} 1 $.
  
An orientable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025080.png" />-dimensional manifold for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025081.png" /> allows a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025082.png" />-imbedding into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025083.png" />. The possibility of immersing an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025084.png" />-dimensional manifold into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025085.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025086.png" /> is related to the Whitney and Pontryagin classes (cf. [[Pontryagin class|Pontryagin class]]) of this manifold. Also, each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025087.png" />-smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025088.png" />-dimensional manifold with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025089.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025090.png" /> allows a proper immersion into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025091.png" /> and a proper imbedding into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025092.png" /> (i.e. an immersion or imbedding such that the pre-image of each compact set is compact). If a [[Riemannian metric|Riemannian metric]] is given on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025093.png" />, one frequently considers an [[Isometric immersion|isometric immersion]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025094.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025095.png" /> or into another Riemannian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025096.png" />. A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025097.png" />-smooth Riemannian manifold (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025098.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i05025099.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i050250100.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i050250101.png" />) allows a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i050250102.png" />-smooth isometric immersion into some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i050250103.png" />. In the case of a compact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i050250104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i050250105.png" />. Conversely, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i050250106.png" />-smooth immersion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i050250107.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i050250108.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i050250109.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i050250110.png" />) induces a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i050250111.png" />-smooth Riemannian metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050250/i050250112.png" /> [[#References|[4]]].
+
An orientable $  m $-
 +
dimensional manifold for $  m \neq 1, 4 $
 +
allows a $  C ^ {l, \alpha } $-
 +
imbedding into $  \mathbf R  ^ {2m-} 1 $.  
 +
The possibility of immersing an $  m $-
 +
dimensional manifold into $  \mathbf R  ^ {n} $
 +
for $  n < 2m- 1 $
 +
is related to the Whitney and Pontryagin classes (cf. [[Pontryagin class|Pontryagin class]]) of this manifold. Also, each $  C ^ {l, \alpha } $-
 +
smooth $  m $-
 +
dimensional manifold with $  l \geq  1 $,
 +
0 \leq  \alpha < 1 $
 +
allows a proper immersion into $  \mathbf R  ^ {2m} $
 +
and a proper imbedding into $  \mathbf R  ^ {2m+} 1 $(
 +
i.e. an immersion or imbedding such that the pre-image of each compact set is compact). If a [[Riemannian metric|Riemannian metric]] is given on $  M  ^ {m} $,  
 +
one frequently considers an [[Isometric immersion|isometric immersion]] of $  M  ^ {m} $
 +
into $  \mathbf R  ^ {n} $
 +
or into another Riemannian space $  N  ^ {n} $.  
 +
A $  C ^ {l, \alpha } $-
 +
smooth Riemannian manifold ( $  l = 2 $,
 +
0 < \alpha < 1 $;  
 +
$  l > 2 $,  
 +
$  0 \leq  \alpha < 1 $)  
 +
allows a $  C ^ {l, \alpha } $-
 +
smooth isometric immersion into some $  \mathbf R  ^ {n} $.  
 +
In the case of a compact $  M  ^ {m} $,  
 +
$  n = ( 2m+ 1)( 6m+ 14) $.  
 +
Conversely, a $  C ^ {l, \alpha } $-
 +
smooth immersion of $  M  ^ {m} $
 +
into $  \mathbf R  ^ {n} $(
 +
$  l \geq  2 $,
 +
0 < \alpha < 1 $)  
 +
induces a $  C ^ {l, \alpha } $-
 +
smooth Riemannian metric on $  M  ^ {m} $[[#References|[4]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Smale,  "The classification of spheres in Euclidean spaces"  ''Ann. of Math.'' , '''69'''  (1959)  pp. 327–344</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Jacobowitz,  "Implicit function theorems and isometric embeddings"  ''Ann. of Math.'' , '''95'''  (1972)  pp. 191–225</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.A. Rokhlin,  D.B. Fuks,  "Beginner's course in topology. Geometric chapters" , Springer  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  I.Kh. Sabitov,  S.Z. Shefel',  "The connections between the order of smoothness of a surface and its metric"  ''Sib. Math. J.'' , '''17''' :  4  (1976)  pp. 687–694  ''Sibirsk. Mat. Zh.'' , '''17''' :  4  (1976)  pp. 916–925</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Smale,  "The classification of spheres in Euclidean spaces"  ''Ann. of Math.'' , '''69'''  (1959)  pp. 327–344</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Jacobowitz,  "Implicit function theorems and isometric embeddings"  ''Ann. of Math.'' , '''95'''  (1972)  pp. 191–225</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.A. Rokhlin,  D.B. Fuks,  "Beginner's course in topology. Geometric chapters" , Springer  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  I.Kh. Sabitov,  S.Z. Shefel',  "The connections between the order of smoothness of a surface and its metric"  ''Sib. Math. J.'' , '''17''' :  4  (1976)  pp. 687–694  ''Sibirsk. Mat. Zh.'' , '''17''' :  4  (1976)  pp. 916–925</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Nash,  "The embedding problem for Riemannian manifolds"  ''Ann. of Math.'' , '''63'''  (1956)  pp. 20–63</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Nash,  "The embedding problem for Riemannian manifolds"  ''Ann. of Math.'' , '''63'''  (1956)  pp. 20–63</TD></TR></table>

Latest revision as of 22:11, 5 June 2020


A continuous mapping $ F: M ^ {m} \rightarrow N ^ {n} $ of an $ m $- dimensional manifold $ M ^ {m} $ into an $ n $- dimensional manifold $ N ^ {n} $ such that for each point $ x \in M ^ {m} $ there exists a neighbourhood $ U _ {x} $ in which $ F $ is an imbedding, i.e. a homeomorphism onto $ F ( U _ {x} ) \subset N ^ {n} $. In particular, if $ F $ is a homeomorphism into $ F ( M ^ {m} ) $, then it is called an imbedding of $ M ^ {m} $ in $ N ^ {n} $. The immersion $ F $ is called a $ C ^ {l, \alpha } $- immersion if $ M ^ {m} $ and $ N ^ {n} $ are $ C ^ {l, \alpha } $-( smooth) manifolds ( $ l \geq 1 $, $ 0 \leq \alpha < 1 $, $ m \leq n $) and if the mapping $ F $ on the corresponding charts is given by functions

$$ x ^ {i} = f ^ { i } ( u ^ {1} \dots u ^ {m} ), \ i = 1 \dots n, $$

that belong to the smoothness class $ C ^ {l, \alpha } $, while the rank of the matrix $ \| df ^ { i } / du ^ {j} \| $ is equal to $ m $ at each point $ x \in M ^ {m} $( a $ C ^ {l, \alpha } $-( smooth) manifold is a manifold provided with a $ \Gamma $- structure, where the pseudo-group consists of mappings that are $ l $- times differentiable and whose derivatives satisfy the Hölder condition of index $ \alpha $).

The concepts of a surface and a $ C ^ {l, \alpha } $-( smooth) surface are closely related to the concepts of an immersion and a $ C ^ {l, \alpha } $-( smooth) immersion. Two immersions $ F $ and $ G $ between manifolds $ M $ and $ N $ are called equivalent if there is a homeomorphism $ \Phi : M \rightarrow M $ such that $ F = G \Phi $.

An immersed manifold is a pair consisting of a manifold $ M $ and an immersion $ F $ of it. A surface of dimension $ m $ in a manifold $ N ^ {n} $ of dimension $ n $ is a class of equivalent immersions $ F: M ^ {m} \rightarrow N ^ {n} $; each immersion of this class is called a parametrization of the surface. A surface is called $ C ^ {l, \alpha } $- smooth if one can introduce $ C ^ {l, \alpha } $- structures in the manifolds $ M $ and $ N $ and if among the parametrizations of the surface one can find a parametrization $ F $ which in these structures is a $ C ^ {l, \alpha } $- immersion.

The theory of immersed manifolds usually deals with properties that are invariant under the above concept of equivalence, and in essence coincides with the theory of surfaces, particularly when one considers topics related to the geometry of immersions.

Let $ M ^ {m} $ be a $ C ^ {l, \alpha } $- manifold, $ l \geq 1 $, $ 0 \leq \alpha < 1 $. Any $ M ^ {m} $ allows for $ m \geq 1 $ an imbedding into the Euclidean space $ \mathbf R ^ {2m} $ and a $ C ^ {l, \alpha } $- immersion into $ \mathbf R ^ {2m-} 1 $ for $ m \geq 2 $. If $ m $ is positive and not a power of $ 2 $, then any $ M ^ {m} $ allows a $ C ^ {l, \alpha } $- imbedding into $ \mathbf R ^ {2m-} 1 $, whereas for any $ m = 2 ^ {s} $ with $ s \geq 0 $ there exist closed smooth $ m $- dimensional manifolds not allowing even a topological imbedding into $ \mathbf R ^ {2m-} 1 $( such as, for example, a projective space). If $ M ^ {m} $ does not have compact components, it allows a $ C ^ {l, \alpha } $- imbedding into $ \mathbf R ^ {2m-} 1 $.

An orientable $ m $- dimensional manifold for $ m \neq 1, 4 $ allows a $ C ^ {l, \alpha } $- imbedding into $ \mathbf R ^ {2m-} 1 $. The possibility of immersing an $ m $- dimensional manifold into $ \mathbf R ^ {n} $ for $ n < 2m- 1 $ is related to the Whitney and Pontryagin classes (cf. Pontryagin class) of this manifold. Also, each $ C ^ {l, \alpha } $- smooth $ m $- dimensional manifold with $ l \geq 1 $, $ 0 \leq \alpha < 1 $ allows a proper immersion into $ \mathbf R ^ {2m} $ and a proper imbedding into $ \mathbf R ^ {2m+} 1 $( i.e. an immersion or imbedding such that the pre-image of each compact set is compact). If a Riemannian metric is given on $ M ^ {m} $, one frequently considers an isometric immersion of $ M ^ {m} $ into $ \mathbf R ^ {n} $ or into another Riemannian space $ N ^ {n} $. A $ C ^ {l, \alpha } $- smooth Riemannian manifold ( $ l = 2 $, $ 0 < \alpha < 1 $; $ l > 2 $, $ 0 \leq \alpha < 1 $) allows a $ C ^ {l, \alpha } $- smooth isometric immersion into some $ \mathbf R ^ {n} $. In the case of a compact $ M ^ {m} $, $ n = ( 2m+ 1)( 6m+ 14) $. Conversely, a $ C ^ {l, \alpha } $- smooth immersion of $ M ^ {m} $ into $ \mathbf R ^ {n} $( $ l \geq 2 $, $ 0 < \alpha < 1 $) induces a $ C ^ {l, \alpha } $- smooth Riemannian metric on $ M ^ {m} $[4].

References

[1] S. Smale, "The classification of spheres in Euclidean spaces" Ann. of Math. , 69 (1959) pp. 327–344
[2] H. Jacobowitz, "Implicit function theorems and isometric embeddings" Ann. of Math. , 95 (1972) pp. 191–225
[3] V.A. Rokhlin, D.B. Fuks, "Beginner's course in topology. Geometric chapters" , Springer (1984) (Translated from Russian)
[4] I.Kh. Sabitov, S.Z. Shefel', "The connections between the order of smoothness of a surface and its metric" Sib. Math. J. , 17 : 4 (1976) pp. 687–694 Sibirsk. Mat. Zh. , 17 : 4 (1976) pp. 916–925

Comments

References

[a1] J. Nash, "The embedding problem for Riemannian manifolds" Ann. of Math. , 63 (1956) pp. 20–63
How to Cite This Entry:
Immersion of a manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Immersion_of_a_manifold&oldid=12393
This article was adapted from an original article by S.Z. Shefel' (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article