This discovery was the real beginning of tensor analysis. In 1869, Christoffel discovered that the components of the intrinsic derivative of a vector field, upon changing the coordinate system, transform as the components of a contravariant vector. Levi-Civita, along with Gregorio Ricci-Curbastro, used Christoffel's symbols to define the notion of parallel transport and explore the relationship of parallel transport with the curvature, thus developing the modern notion of holonomy. The Levi-Civita connection is named after Tullio Levi-Civita, although originally "discovered" by Elwin Bruno Christoffel. The components (structure coefficients) of this connection with respect to a system of local coordinates are called Christoffel symbols.
In the theory of Riemannian and pseudo-Riemannian manifolds the term covariant derivative is often used for the Levi-Civita connection.
The fundamental theorem of Riemannian geometry states that there is a unique connection which satisfies these properties. affine connection) that preserves the ( pseudo-) Riemannian metric and is torsion-free. In Riemannian or pseudo-Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold (i.e. Affine connection on the tangent bundle of a manifold
0 kommentar(er)
