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| CATEGORIES ABOUT AFFINE CONNECTION | |
| differential geometry | |
| connection mathematics | |
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The curvature of a connected manifold can be characterised intrinsically by taking a vector at some point and Parallel Transporting it along a curve on the manifold. Although comparing vectors at different points is generally not a well-defined process, an affine connection is a rule which describes how to legitimately move a vector along a curve on the manifold without changing its direction ('keeping the vector parallel'). The affine connection is a linear map sending two vector fields ( and ) to a third one. If this vector field vanishes, is said to be parallel transported along - this expresses the desired property of the vector pointing in the same direction along the integral curves of . The affine connection thus permits a notion of parallel transport to be defined. The Torsion of a connection is a smooth vector field. If the torsion vanishes, the connection is said to be ''torsion-free'' or more commonly, ''symmetric''. An important affine connection is the Levi-Civita Connection , which is a symmetric connection that results from parallel transporting a tangent vector along a curve whilst keeping the inner product of that vector constant along the curve. The resulting connection coefficients are called Christoffel Symbols and can be calculated directly from the metric. For this reason, this type of connection is often called a ''metric connection''. |
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