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advanced_tools:connections:levi_civita_connection

$ \Gamma_{ijk} \equiv -\bigl(\partial_{i}g_{jk}-\partial_{k}g_{ij}-\partial_{j}g_{ki}\bigr)$

*also known as Christoffel Symbols; see also Connections *

The Levi-Civita connection is a mathematical tool that we use to parallel transport vectors around a manifold.

Parallel transport is just the simplest way to compare vectors at different points in the manifold.

Parallel is necessary, for example, to define the covariant derivative.

Christoffel symbols $\Gamma^i_{jk}$ are a particular type of connection that a Lorentzian manifold has (called the Levi-Civita connection).

**Examples**

The diagram below shows three concrete examples for connections (Christoffel symbols) on simple 2-dimensional manifolds. For a more detailed explanation see Fun with Symmetry.

The motto in this section is: *the higher the level of abstraction, the better*.

The Christoffel symbols appear in the most important equations of general relativity: the Einstein equation and the geodesic equation.

advanced_tools/connections/levi_civita_connection.txt · Last modified: 2021/08/23 02:30 by edi

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