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A mapping of the tangent space of a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e0369301.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e0369302.png" />. It is defined by a [[Connection|connection]] given on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e0369303.png" /> and is a far-reaching generalization of the ordinary exponential function regarded as a mapping of a straight line into itself.
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1) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e0369304.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e0369305.png" />-manifold with an affine connection, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e0369306.png" /> be a point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e0369307.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e0369308.png" /> be the tangent space to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e0369309.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693010.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693011.png" /> be a non-zero vector in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693012.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693013.png" /> be the geodesic passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693014.png" /> in the direction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693015.png" />. There is an open neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693016.png" /> of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693017.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693018.png" /> and an open neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693019.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693020.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693021.png" /> such that the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693022.png" /> is a diffeomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693023.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693024.png" />. This mapping is called the exponential mapping at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693025.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693026.png" />. A neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693027.png" /> is called normal if: 1) the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693028.png" /> maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693029.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693030.png" /> diffeomorphically; and 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693032.png" /> imply that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693033.png" />. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693034.png" /> is said to be a normal neighbourhood of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693035.png" /> in the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693036.png" />. Every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693037.png" /> has a convex normal neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693038.png" />: Any two points of such a neighbourhood can be joined by exactly one geodesic segment lying in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693039.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693040.png" /> is a complete Riemannian manifold, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693041.png" /> is a surjective mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693042.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693043.png" />.
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2) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693044.png" /> be a Lie group with identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693045.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693046.png" /> be the corresponding Lie algebra consisting of the tangent vectors to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693047.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693048.png" />. For every vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693049.png" /> there is a unique differentiable homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693050.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693051.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693052.png" /> such that the tangent vector to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693053.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693054.png" /> coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693055.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693056.png" /> is called the exponential mapping of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693057.png" /> into the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693058.png" />. There is an open neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693059.png" /> of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693060.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693061.png" /> and an open neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693062.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693063.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693064.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693065.png" /> is a diffeomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693066.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693067.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693068.png" /> be some basis for the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693069.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693070.png" /> is a coordinate system on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693071.png" />; these coordinates are called canonical.
+
A mapping of the tangent space of a manifold  $  M $
 +
into $  M $.  
 +
It is defined by a [[Connection|connection]] given on  $  M $
 +
and is a far-reaching generalization of the ordinary exponential function regarded as a mapping of a straight line into itself.
  
The concept of an exponential mapping of a Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693072.png" /> can also be approached from another point of view. There is a one-to-one correspondence between the set of all affine connections on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693073.png" /> that are invariant relative to the group of left translations and the set of bilinear functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693074.png" />. It turns out that the exponential mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693075.png" /> of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693076.png" /> into the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693077.png" /> coincides with the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693078.png" /> of the tangent space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693079.png" /> into the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693080.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693081.png" /> in this manifold with respect to the left-invariant affine connection corresponding to any skew-symmetric bilinear function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036930/e03693082.png" />.
+
1) Let  $  M $
 +
be a  $  C  ^  \infty  $-
 +
manifold with an affine connection, let  $  p $
 +
be a point in  $  M $,
 +
let  $  M _ {p} $
 +
be the tangent space to  $  M $
 +
at  $  p $,
 +
let  $  X $
 +
be a non-zero vector in  $  M _ {p} $,
 +
and let  $  t \rightarrow \gamma _ {X} ( t) $
 +
be the geodesic passing through  $  p $
 +
in the direction of $  X $.
 +
There is an open neighbourhood  $  N _ {0} $
 +
of the point  $  0 $
 +
in  $  M _ {p} $
 +
and an open neighbourhood  $  N _ {p} $
 +
of  $  p $
 +
in  $  M $
 +
such that the mapping  $  X \rightarrow \gamma _ {X} ( 1) $
 +
is a diffeomorphism of  $  N _ {0} $
 +
onto  $  N _ {p} $.
 +
This mapping is called the exponential mapping at  $  p $
 +
and is denoted by  $  \mathop{\rm exp} $.
 +
A neighbourhood  $  N _ {0} $
 +
is called normal if: 1) the mapping  $  \mathop{\rm exp} $
 +
maps  $  N _ {0} $
 +
onto  $  N _ {p} $
 +
diffeomorphically; and 2)  $  X \in N _ {0} $
 +
and  $  0 \leq  t \leq  1 $
 +
imply that  $  t X \in N _ {0} $.
 +
In this case  $  N _ {p} $
 +
is said to be a normal neighbourhood of the point  $  p $
 +
in the manifold  $  M $.
 +
Every  $  p \in M $
 +
has a convex normal neighbourhood  $  N _ {p} $:
 +
Any two points of such a neighbourhood can be joined by exactly one geodesic segment lying in  $  N _ {p} $.
 +
If  $  M $
 +
is a complete Riemannian manifold, then  $  \mathop{\rm exp} $
 +
is a surjective mapping of $  M _ {p} $
 +
onto  $  M $.
 +
 
 +
2) Let  $  G $
 +
be a Lie group with identity  $  e $
 +
and let  $  \mathfrak g $
 +
be the corresponding Lie algebra consisting of the tangent vectors to  $  G $
 +
at  $  e $.
 +
For every vector  $  X \in \mathfrak g $
 +
there is a unique differentiable homomorphism  $  \theta $
 +
of the group  $  \mathbf R $
 +
into  $  G $
 +
such that the tangent vector to  $  \theta ( \mathbf R ) $
 +
at  $  e $
 +
coincides with  $  X $.
 +
The mapping  $  X \rightarrow  \mathop{\rm exp}  X = \theta ( 1) $
 +
is called the exponential mapping of the algebra  $  \mathfrak g $
 +
into the group  $  G $.
 +
There is an open neighbourhood  $  N _ {0} $
 +
of the point  $  0 $
 +
in  $  \mathfrak g $
 +
and an open neighbourhood  $  N _ {e} $
 +
of  $  e $
 +
in  $  G $
 +
such that  $  \mathop{\rm exp} $
 +
is a diffeomorphism of  $  N _ {0} $
 +
onto  $  N _ {e} $.  
 +
Let  $  X _ {1} \dots X _ {n} $
 +
be some basis for the algebra  $  \mathfrak g $.  
 +
The mapping  $  \mathop{\rm exp}  ( x _ {1} X _ {1} + {} \dots + x _ {n} X _ {n} ) \rightarrow ( x _ {1} \dots x _ {n} ) $
 +
is a coordinate system on  $  N _ {e} $;
 +
these coordinates are called canonical.
 +
 
 +
The concept of an exponential mapping of a Lie group  $  G $
 +
can also be approached from another point of view. There is a one-to-one correspondence between the set of all affine connections on $  G $
 +
that are invariant relative to the group of left translations and the set of bilinear functions $  \alpha : \mathfrak g \times \mathfrak g \rightarrow \mathfrak g $.  
 +
It turns out that the exponential mapping $  \mathop{\rm exp} $
 +
of the algebra $  \mathfrak g $
 +
into the group $  G $
 +
coincides with the mapping $  \mathop{\rm exp} $
 +
of the tangent space of $  \mathfrak g $
 +
into the manifold $  G $
 +
at the point e $
 +
in this manifold with respect to the left-invariant affine connection corresponding to any skew-symmetric bilinear function $  \alpha $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Helgason,  "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press  (1978)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Helgason,  "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press  (1978)</TD></TR></table>

Latest revision as of 19:38, 5 June 2020


A mapping of the tangent space of a manifold $ M $ into $ M $. It is defined by a connection given on $ M $ and is a far-reaching generalization of the ordinary exponential function regarded as a mapping of a straight line into itself.

1) Let $ M $ be a $ C ^ \infty $- manifold with an affine connection, let $ p $ be a point in $ M $, let $ M _ {p} $ be the tangent space to $ M $ at $ p $, let $ X $ be a non-zero vector in $ M _ {p} $, and let $ t \rightarrow \gamma _ {X} ( t) $ be the geodesic passing through $ p $ in the direction of $ X $. There is an open neighbourhood $ N _ {0} $ of the point $ 0 $ in $ M _ {p} $ and an open neighbourhood $ N _ {p} $ of $ p $ in $ M $ such that the mapping $ X \rightarrow \gamma _ {X} ( 1) $ is a diffeomorphism of $ N _ {0} $ onto $ N _ {p} $. This mapping is called the exponential mapping at $ p $ and is denoted by $ \mathop{\rm exp} $. A neighbourhood $ N _ {0} $ is called normal if: 1) the mapping $ \mathop{\rm exp} $ maps $ N _ {0} $ onto $ N _ {p} $ diffeomorphically; and 2) $ X \in N _ {0} $ and $ 0 \leq t \leq 1 $ imply that $ t X \in N _ {0} $. In this case $ N _ {p} $ is said to be a normal neighbourhood of the point $ p $ in the manifold $ M $. Every $ p \in M $ has a convex normal neighbourhood $ N _ {p} $: Any two points of such a neighbourhood can be joined by exactly one geodesic segment lying in $ N _ {p} $. If $ M $ is a complete Riemannian manifold, then $ \mathop{\rm exp} $ is a surjective mapping of $ M _ {p} $ onto $ M $.

2) Let $ G $ be a Lie group with identity $ e $ and let $ \mathfrak g $ be the corresponding Lie algebra consisting of the tangent vectors to $ G $ at $ e $. For every vector $ X \in \mathfrak g $ there is a unique differentiable homomorphism $ \theta $ of the group $ \mathbf R $ into $ G $ such that the tangent vector to $ \theta ( \mathbf R ) $ at $ e $ coincides with $ X $. The mapping $ X \rightarrow \mathop{\rm exp} X = \theta ( 1) $ is called the exponential mapping of the algebra $ \mathfrak g $ into the group $ G $. There is an open neighbourhood $ N _ {0} $ of the point $ 0 $ in $ \mathfrak g $ and an open neighbourhood $ N _ {e} $ of $ e $ in $ G $ such that $ \mathop{\rm exp} $ is a diffeomorphism of $ N _ {0} $ onto $ N _ {e} $. Let $ X _ {1} \dots X _ {n} $ be some basis for the algebra $ \mathfrak g $. The mapping $ \mathop{\rm exp} ( x _ {1} X _ {1} + {} \dots + x _ {n} X _ {n} ) \rightarrow ( x _ {1} \dots x _ {n} ) $ is a coordinate system on $ N _ {e} $; these coordinates are called canonical.

The concept of an exponential mapping of a Lie group $ G $ can also be approached from another point of view. There is a one-to-one correspondence between the set of all affine connections on $ G $ that are invariant relative to the group of left translations and the set of bilinear functions $ \alpha : \mathfrak g \times \mathfrak g \rightarrow \mathfrak g $. It turns out that the exponential mapping $ \mathop{\rm exp} $ of the algebra $ \mathfrak g $ into the group $ G $ coincides with the mapping $ \mathop{\rm exp} $ of the tangent space of $ \mathfrak g $ into the manifold $ G $ at the point $ e $ in this manifold with respect to the left-invariant affine connection corresponding to any skew-symmetric bilinear function $ \alpha $.

References

[1] S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978)
How to Cite This Entry:
Exponential mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Exponential_mapping&oldid=46876
This article was adapted from an original article by A.S. Fedenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article