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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.

====Comments====
The Euclidean connection is also sometimes called the [[Metric connection|metric connection]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Kobayashi,   K. Nomizu,   "Foundations of differential geometry" , '''1''' , Interscience  (1963)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Klingenberg,   "Riemannian geometry" , de Gruyter  (1982)  pp. Chapt. 1.8  (Translated from German)</TD></TR></table>