Euclidean connection

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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.


The Euclidean connection is also sometimes called the metric connection.


[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)
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Euclidean connection. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by Ü. Lumiste (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article