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| The invariant interval between two points on a Riemann manifold is | |
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Let  be
the co-variant derivative, the invariance of ds, requires |
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| For the affine connection to be determined by a metric tensor only, two cases arise | |
| Case I: The metric and affine connection are both symmetric | |
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| With the conditions given by equation 1.2, and 1.3, the symmetric affine connection are the Christoffel Symbols, see reference [1] | |
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| Case 2: The metric and affine connection are both asymmetric | |
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| With the conditions given by equation 1.2, and 1.5, the asymmetric affine connection is | |
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| A general affine connection can be formed from equations 1.4 and 1.6 | |
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where the imaginary part of the
connection is asymmetric in  and 
It can be shown that using condition 1.2 with the connection 1.7, that the affine connection is |
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| where | |
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| the asymmetric affine connection is completely asymmetric |