WeylConnection

A Weyl connection is a mathematical concept in the field of differential geometry, particularly in the context of general relativity and gauge theories. It is a generalization of the standard Levi-Civita connection, which is used to define parallel transport, covariant derivatives, and curvature on a manifold.

Weyl connections are named after Hermann Weyl, a German mathematician and theoretical physicist who introduced the concept in the early 20th century. He was interested in developing a unified theory of electromagnetism and gravity, and the Weyl connection was part of his attempt to achieve this goal.

A Weyl connection is defined by the following formula:

Γₖᵢⱼ = {ₖᵢⱼ} + gₖᵢΦⱼ - gₖⱼΦᵢ

where Γₖᵢⱼ is the Weyl connection, {ₖᵢⱼ} is the Levi-Civita connection (Christoffel symbols of the second kind), gₖᵢ is the metric tensor, and Φⱼ and Φᵢ are vector fields. The Weyl connection is characterized by the fact that it allows the parallel transport of vectors without preserving their lengths or angles, which means it is not a metric connection.

The concept of a Weyl connection is important in the study of conformal geometry, where it is used to define a so-called Weyl curvature tensor. This tensor is conformally invariant, which means it remains unchanged under conformal transformations of the metric tensor. In general relativity, Weyl connections have been used to study various aspects of gravity and spacetime structure, such as the propagation of light and the behavior of particles in curved spacetimes.