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- Fundamental_theorem_of_Riemannian_geometry abstract "In Riemannian geometry, the fundamental theorem of Riemannian geometry states that on any Riemannian manifold (or pseudo-Riemannian manifold) there is a unique torsion-free metric connection, called the Levi-Civita connection of the given metric. Here a metric (or Riemannian) connection is a connection which preserves the metric tensor. More precisely:Fundamental Theorem of Riemannian Geometry. Let (M, g) be a Riemannian manifold (or pseudo-Riemannian manifold). Then there is a unique connection ∇ which satisfies the following conditions:for any vector fields X, Y, Z we have where denotes the derivative of the function along vector field X.for any vector fields X, Y, where [X, Y] denotes the Lie bracket for vector fields X, Y. The first condition means that the metric tensor is preserved by parallel transport, while the second condition expresses the fact that the torsion of ∇ is zero.An extension of the fundamental theorem states that given a pseudo-Riemannian manifold there is a unique connection preserving the metric tensor with any given vector-valued 2-form as its torsion.The following technical proof presents a formula for Christoffel symbols of the connection in a local coordinate system. For a given metric this set of equations can become rather complicated. There are quicker and simpler methods to obtain the Christoffel symbols for a given metric, e.g. using the action integral and the associated Euler-Lagrange equations.".
- Fundamental_theorem_of_Riemannian_geometry wikiPageID "845060".
- Fundamental_theorem_of_Riemannian_geometry wikiPageRevisionID "582505272".
- Fundamental_theorem_of_Riemannian_geometry hasPhotoCollection Fundamental_theorem_of_Riemannian_geometry.
- Fundamental_theorem_of_Riemannian_geometry subject Category:Articles_containing_proofs.
- Fundamental_theorem_of_Riemannian_geometry subject Category:Connection_(mathematics).
- Fundamental_theorem_of_Riemannian_geometry subject Category:Fundamental_theorems.
- Fundamental_theorem_of_Riemannian_geometry subject Category:Theorems_in_Riemannian_geometry.
- Fundamental_theorem_of_Riemannian_geometry type Abstraction100002137.
- Fundamental_theorem_of_Riemannian_geometry type Communication100033020.
- Fundamental_theorem_of_Riemannian_geometry type FundamentalTheorems.
- Fundamental_theorem_of_Riemannian_geometry type Message106598915.
- Fundamental_theorem_of_Riemannian_geometry type Proposition106750804.
- Fundamental_theorem_of_Riemannian_geometry type Statement106722453.
- Fundamental_theorem_of_Riemannian_geometry type Theorem106752293.
- Fundamental_theorem_of_Riemannian_geometry type TheoremsInGeometry.
- Fundamental_theorem_of_Riemannian_geometry type TheoremsInRiemannianGeometry.
- Fundamental_theorem_of_Riemannian_geometry comment "In Riemannian geometry, the fundamental theorem of Riemannian geometry states that on any Riemannian manifold (or pseudo-Riemannian manifold) there is a unique torsion-free metric connection, called the Levi-Civita connection of the given metric. Here a metric (or Riemannian) connection is a connection which preserves the metric tensor. More precisely:Fundamental Theorem of Riemannian Geometry. Let (M, g) be a Riemannian manifold (or pseudo-Riemannian manifold).".
- Fundamental_theorem_of_Riemannian_geometry label "Fundamental theorem of Riemannian geometry".
- Fundamental_theorem_of_Riemannian_geometry label "Teorema fundamental de la geometría de Riemann".
- Fundamental_theorem_of_Riemannian_geometry sameAs Teorema_fundamental_de_la_geometría_de_Riemann.
- Fundamental_theorem_of_Riemannian_geometry sameAs Hoofdstelling_van_de_riemann-meetkunde.
- Fundamental_theorem_of_Riemannian_geometry sameAs m.03ggsn.
- Fundamental_theorem_of_Riemannian_geometry sameAs Q2185349.
- Fundamental_theorem_of_Riemannian_geometry sameAs Q2185349.
- Fundamental_theorem_of_Riemannian_geometry sameAs Fundamental_theorem_of_Riemannian_geometry.
- Fundamental_theorem_of_Riemannian_geometry wasDerivedFrom Fundamental_theorem_of_Riemannian_geometry?oldid=582505272.
- Fundamental_theorem_of_Riemannian_geometry isPrimaryTopicOf Fundamental_theorem_of_Riemannian_geometry.