advanced_tools:connections:levi_civita_connection
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advanced_tools:connections:levi_civita_connection [2018/04/14 11:22] aresmarrero ↷ Page moved from advanced_notions:general_relativity:christoffel_symbols to advanced_tools:connections:christoffel_symbols |
advanced_tools:connections:levi_civita_connection [2025/03/04 00:55] (current) edi [Concrete] |
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| <WRAP lag>$ \Gamma_{ijk} \equiv -\bigl(\partial_{i}g_{jk}-\partial_{k}g_{ij}-\partial_{j}g_{ki}\bigr)$</WRAP> | <WRAP lag>$ \Gamma_{ijk} \equiv -\bigl(\partial_{i}g_{jk}-\partial_{k}g_{ij}-\partial_{j}g_{ki}\bigr)$</WRAP> | ||
| - | ====== Levi-Civita connection ====== | + | ====== Levi-Civita Connection ====== |
| //also known as Christoffel Symbols; see also [[advanced_tools:connections]] // | //also known as Christoffel Symbols; see also [[advanced_tools:connections]] // | ||
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| <tabbox Intuitive> | <tabbox Intuitive> | ||
| - | The Levi-Civita connection is a mathematical tool that we use to parallel transport vectors around a manifold. | + | The Levi-Civita connection is a mathematical tool that we use to [[advanced_tools:parallel_transport|parallel transport]] vectors around a manifold. |
| Parallel transport is just the simplest way to compare vectors at different points in the manifold. | Parallel transport is just the simplest way to compare vectors at different points in the manifold. | ||
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| Parallel is necessary, for example, to define the covariant derivative. | Parallel is necessary, for example, to define the covariant derivative. | ||
| <tabbox Concrete> | <tabbox Concrete> | ||
| - | Christoffel symbols $\Gamma^i_{jk}$ are a particular type of connection that a Lorentzian manifold has (called the Levi-Civita connection). | + | Christoffel symbols $\Gamma^i_{jk}$ are a particular type of connection that a Lorentzian manifold has (called the Levi-Civita connection). |
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| + | ---- | ||
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| + | **Examples** | ||
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| + | The diagram below shows three concrete examples for connections (Christoffel symbols) on simple 2-dimensional manifolds. For a more detailed explanation see [[https://esackinger.wordpress.com/appendices/#riemannian_geometry|Fun with Symmetry]]. | ||
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| + | {{:advanced_tools:metric_connect_curvature.jpg?nolink}} | ||
| <tabbox Abstract> | <tabbox Abstract> | ||
advanced_tools/connections/levi_civita_connection.1523697739.txt.gz · Last modified: 2018/04/14 09:22 (external edit)
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