Chapter VII

Linear Connections

Keywords

Linear connections on manifolds; Christoffel symbols; Covariant derivatives; Geodesics; Torsion tensor; Curvature tensor; Cartan connection; Moving frames; Bianchi identities; Levi-Civita connection; Ricci tensor; Differential operators

7.1 Scope of the Chapter

Derivatives of components of vector and tensor fields on differentiable manifolds with respect to local coordinates are not generally components of a tensor. However, it becomes possible to define a sort of derivative of a tensor field which proves to be also a tensor through a geometrical structure imposed on a manifold called a linear or affine connection. In this way, we can properly accomplish the task of extending the classical differential geometry ...

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