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Can define parallel vector fields as \(\nabla X = 0\), a PDE.
- Don’t generally exist, this is an overdetermined equation. The integrability condition for this equation is equivalent to \(\mathop{\mathrm{Curv}}(\nabla) = 0\).
- If curvature vanishes, parallel transport along every curve can be used to define parallel vector fields on \(M\).
curvature of a connection
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curvature
Topics: - Riemannian geometry - Physics - connection - Riemann curvature - Ricci curvature - Gaussian curvature - scalar curvature - Sectional curvature - curvature of a connection
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connection
The main invariants of an affine connection are its curvature of a connection