Euclidean connection

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)