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Difference between revisions of "Euclidean connection"

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A differential-geometric structure on a Euclidean vector bundle, generalizing the [[Levi-Civita connection|Levi-Civita connection]], or [[Riemannian connection|Riemannian connection]], in Riemannian geometry. A smooth vector bundle is called Euclidean if each of its fibres has the structure of a Euclidean vector space with a scalar product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036330/e0363301.png" /> such that for any smooth sections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036330/e0363302.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036330/e0363303.png" /> the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036330/e0363304.png" /> is a smooth function on the base. A linear connection on a Euclidean vector bundle is called a Euclidean connection if for any parallel displacement of two vectors their scalar product remains constant. This is equivalent to the metric tensor defining the scalar product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036330/e0363305.png" /> in each fibre being covariantly constant. The Euclidean connection in the tangent bundle of a Riemannian space is the Riemannian connection. Sometimes the term  "Euclidean connection"  is used only in this case, and then  "Riemannian connection"  means the Levi-Civita connection.
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A differential-geometric structure on a Euclidean vector bundle, generalizing the [[Levi-Civita connection|Levi-Civita connection]], or [[Riemannian connection|Riemannian connection]], in Riemannian geometry. A smooth vector bundle is called Euclidean if each of its fibres has the structure of a Euclidean vector space with a scalar product  $  \langle  , \rangle $
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such that for any smooth sections  $  X $
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and  $  Y $
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the function  $  \langle  X , Y \rangle $
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is a smooth function on the base. A linear connection on a Euclidean vector bundle is called a Euclidean connection if for any parallel displacement of two vectors their scalar product remains constant. This is equivalent to the metric tensor defining the scalar product  $  \langle  , \rangle $
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in each fibre being covariantly constant. The Euclidean connection in the tangent bundle of a Riemannian space is the Riemannian connection. Sometimes the term  "Euclidean connection"  is used only in this case, and then  "Riemannian connection"  means the Levi-Civita connection.
  
 
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Latest revision as of 19:38, 5 June 2020


A differential-geometric structure on a Euclidean vector bundle, generalizing the Levi-Civita connection, or Riemannian connection, in Riemannian geometry. A smooth vector bundle is called Euclidean if each of its fibres has the structure of a Euclidean vector space with a scalar product $ \langle , \rangle $ such that for any smooth sections $ X $ and $ Y $ the function $ \langle X , Y \rangle $ is a smooth function on the base. A linear connection on a Euclidean vector bundle is called a Euclidean connection if for any parallel displacement of two vectors their scalar product remains constant. This is equivalent to the metric tensor defining the scalar product $ \langle , \rangle $ in each fibre being covariantly constant. The Euclidean connection in the tangent bundle of a Riemannian space is the Riemannian connection. Sometimes the term "Euclidean connection" is used only in this case, and then "Riemannian connection" means the Levi-Civita connection.

Comments

The Euclidean connection is also sometimes called the metric connection.

References

[a1] S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1 , Interscience (1963)
[a2] W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) pp. Chapt. 1.8 (Translated from German)
How to Cite This Entry:
Euclidean connection. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euclidean_connection&oldid=46855
This article was adapted from an original article by Ü. Lumiste (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article