# 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) |

**How to Cite This Entry:**

Euclidean connection.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Euclidean_connection&oldid=46855