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''of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092370/t0923702.png" /> on a differentiable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092370/t0923703.png" />''
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The [[Vector bundle|vector bundle]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092370/t0923704.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092370/t0923705.png" /> associated with the bundle of tangent frames and having as standard fibre the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092370/t0923706.png" /> of tensors (cf. [[Tensor on a vector space|Tensor on a vector space]]) of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092370/t0923707.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092370/t0923708.png" />, on which the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092370/t0923709.png" /> acts by the tensor representation. For instance, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092370/t09237010.png" /> coincides with the [[Tangent bundle|tangent bundle]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092370/t09237011.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092370/t09237012.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092370/t09237013.png" /> coincides with the cotangent bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092370/t09237014.png" />. In the general case, the tensor bundle is isomorphic to the tensor product of the tangent and cotangent bundles:
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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/t/t092/t092370/t09237015.png" /></td> </tr></table>
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''of type  $  ( p, q) $
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on a differentiable manifold  $  M $''
  
Sections of the tensor bundle of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092370/t09237016.png" /> are called tensor fields of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092370/t09237017.png" /> and are the basic object of study in differential geometry. For example, a Riemannian structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092370/t09237018.png" /> is a smooth section of the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092370/t09237019.png" /> the values of which are positive-definite symmetric forms. The smooth sections of the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092370/t09237020.png" /> form a module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092370/t09237021.png" /> over the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092370/t09237022.png" /> of smooth functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092370/t09237023.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092370/t09237024.png" /> is a paracompact Hausdorff manifold, then
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The [[Vector bundle|vector bundle]]  $  T  ^ {p,q} ( M) $
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over  $  M $
 +
associated with the bundle of tangent frames and having as standard fibre the space  $  T  ^ {p,q} ( \mathbf R  ^ {n} ) $
 +
of tensors (cf. [[Tensor on a vector space|Tensor on a vector space]]) of type $  ( p, q) $
 +
on  $  \mathbf R  ^ {n} $,
 +
on which the group  $  \mathop{\rm GL} ( n, \mathbf R ) $
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acts by the tensor representation. For instance,  $  T  ^ {1,0} ( M) $
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coincides with the [[Tangent bundle|tangent bundle]]  $  T M $
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over  $  M $,
 +
while  $  T  ^ {0,1} ( M) $
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coincides with the cotangent bundle  $  T  ^ {*} M $.  
 +
In the general case, the tensor bundle is isomorphic to the tensor product of the tangent and cotangent bundles:
  
<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/t/t092/t092370/t09237025.png" /></td> </tr></table>
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$$
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T  ^ {p,q} ( M)  \cong  \otimes ^ { p }  TM \otimes \otimes ^ { q }  T  ^ {*} M .
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092370/t09237026.png" /> is the module of smooth vector fields, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092370/t09237027.png" /> is the module of Pfaffian differential forms (cf. also [[Pfaffian form|Pfaffian form]]), and the tensor products are taken over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092370/t09237028.png" />. In classical differential geometry tensor fields are sometimes simply called tensors on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092370/t09237029.png" />.
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Sections of the tensor bundle of type  $  ( p, q) $
 +
are called tensor fields of type  $  ( p, q) $
 +
and are the basic object of study in differential geometry. For example, a Riemannian structure on  $  M $
 +
is a smooth section of the bundle  $  T  ^ {0,2} ( M) $
 +
the values of which are positive-definite symmetric forms. The smooth sections of the bundle  $  T  ^ {p,q} ( M) $
 +
form a module  $  D  ^ {p,q} ( M) $
 +
over the algebra  $  F ^ { \infty } ( M) = D  ^ {0,0} ( M) $
 +
of smooth functions on  $  M $.
 +
If  $  M $
 +
is a paracompact Hausdorff manifold, then
 +
 
 +
$$
 +
D  ^ {p,q} ( M)  \cong \
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\otimes ^ { p }  D  ^ {1} ( M) \otimes
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\otimes ^ { q }  D  ^ {1} ( M)  ^ {*} ,
 +
$$
 +
 
 +
where $  D  ^ {1} ( M) = D  ^ {1,0} ( M) $
 +
is the module of smooth vector fields, $  D  ^ {1} ( M)  ^ {*} = D  ^ {0,1} ( M) $
 +
is the module of Pfaffian differential forms (cf. also [[Pfaffian form|Pfaffian form]]), and the tensor products are taken over $  F ^ { \infty } ( M) $.  
 +
In classical differential geometry tensor fields are sometimes simply called tensors on $  M $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Kobayashi,  K. Nomizu,  "Foundations of differential geometry" , '''1''' , Interscience  (1963)</TD></TR><TR><TD valign="top">[2]</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. Kobayashi,  K. Nomizu,  "Foundations of differential geometry" , '''1''' , Interscience  (1963)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Helgason,  "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press  (1978)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092370/t09237030.png" /> of vector fields is often denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092370/t09237031.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092370/t09237032.png" />, the space of Pfaffian forms, by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092370/t09237033.png" />.
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The space $  D  ^ {1} ( M) $
 +
of vector fields is often denoted by $  X( M) $,  
 +
and $  D  ^ {1} ( M)  ^ {*} $,  
 +
the space of Pfaffian forms, by $  \Omega  ^ {1} ( M) $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Klingenberg,  "Riemannian geometry" , de Gruyter  (1982)  (Translated from German)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Klingenberg,  "Riemannian geometry" , de Gruyter  (1982)  (Translated from German)</TD></TR></table>

Latest revision as of 08:25, 6 June 2020


of type $ ( p, q) $ on a differentiable manifold $ M $

The vector bundle $ T ^ {p,q} ( M) $ over $ M $ associated with the bundle of tangent frames and having as standard fibre the space $ T ^ {p,q} ( \mathbf R ^ {n} ) $ of tensors (cf. Tensor on a vector space) of type $ ( p, q) $ on $ \mathbf R ^ {n} $, on which the group $ \mathop{\rm GL} ( n, \mathbf R ) $ acts by the tensor representation. For instance, $ T ^ {1,0} ( M) $ coincides with the tangent bundle $ T M $ over $ M $, while $ T ^ {0,1} ( M) $ coincides with the cotangent bundle $ T ^ {*} M $. In the general case, the tensor bundle is isomorphic to the tensor product of the tangent and cotangent bundles:

$$ T ^ {p,q} ( M) \cong \otimes ^ { p } TM \otimes \otimes ^ { q } T ^ {*} M . $$

Sections of the tensor bundle of type $ ( p, q) $ are called tensor fields of type $ ( p, q) $ and are the basic object of study in differential geometry. For example, a Riemannian structure on $ M $ is a smooth section of the bundle $ T ^ {0,2} ( M) $ the values of which are positive-definite symmetric forms. The smooth sections of the bundle $ T ^ {p,q} ( M) $ form a module $ D ^ {p,q} ( M) $ over the algebra $ F ^ { \infty } ( M) = D ^ {0,0} ( M) $ of smooth functions on $ M $. If $ M $ is a paracompact Hausdorff manifold, then

$$ D ^ {p,q} ( M) \cong \ \otimes ^ { p } D ^ {1} ( M) \otimes \otimes ^ { q } D ^ {1} ( M) ^ {*} , $$

where $ D ^ {1} ( M) = D ^ {1,0} ( M) $ is the module of smooth vector fields, $ D ^ {1} ( M) ^ {*} = D ^ {0,1} ( M) $ is the module of Pfaffian differential forms (cf. also Pfaffian form), and the tensor products are taken over $ F ^ { \infty } ( M) $. In classical differential geometry tensor fields are sometimes simply called tensors on $ M $.

References

[1] S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1 , Interscience (1963)
[2] S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978)

Comments

The space $ D ^ {1} ( M) $ of vector fields is often denoted by $ X( M) $, and $ D ^ {1} ( M) ^ {*} $, the space of Pfaffian forms, by $ \Omega ^ {1} ( M) $.

References

[a1] W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German)
How to Cite This Entry:
Tensor bundle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tensor_bundle&oldid=17758
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article