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 $ \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. | ||
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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) |
Euclidean connection. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euclidean_connection&oldid=12022