Levi-Civita Connection

In Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold. More specifically, it is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle (an affine connection) preserving a given (pseudo-)Riemannian metric.

The fundamental theorem of Riemannian geometry states that there is a unique connection which satisfies these properties.

In the theory of Riemannian and pseudo-Riemannian manifolds the term covariant derivative is often used for the Levi-Civita connection. The components of this connection with respect to a system of local coordinates are called Christoffel symbols.

The Levi-Civita connection is named after Tullio Levi-Civita, although originally "discovered" by Elwin Bruno Christoffel. 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 notions of intrinsic derivative and parallel displacement of a vector along a curve make sense on an abstract Riemannian manifold, even though the original motivation relied on a specific embedding, since the definition of the Christoffel symbols make sense in any Riemannian manifold. In 1869, Christoffel discovered that the components of the intrinsic derivative of a vector transform as the components of a contravariant vector. This discovery was the real beginning of tensor analysis. It was not until 1917 that Levi-Civita interpreted the intrinsic derivative in the case of an embedded surface as the tangential component of the usual derivative in the ambient affine space.

Read more about Levi-Civita Connection:  Formal Definition, Christoffel Symbols, Derivative Along Curve, Parallel Transport, Example: The Unit Sphere in ℝ3

Other articles related to "connection":

Levi-Civita Connection - Example: The Unit Sphere in ℝ3
... Lemma The formula defines an affine connection on S2 with vanishing torsion ... It is also a straightforward computation to show that this connection is torsion free ... In fact, this connection is the Levi-Civita connection for the metric on S2 inherited from R3 ...

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