Retrieving "Homology" from the archives
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Riemannian Manifold
Linked via "homology"
Parallel transport is the process of moving a vector field along a curve such that its covariant derivative along the curve is zero, effectively preserving the vector relative to the local metric structure. The set of all linear transformations generated by parallel transporting vectors around all possible closed loops based at a point $p$ forms the Holonomy Group, denoted $\text{Hol}(M, p)$. The structure of the holonomy group re…
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Torsion Free Module
Linked via "homology"
$$ \text{rank}(M) = \dimQ (M \otimesR Q) $$
The rank reflects the "size" of the module in terms of independent components relative to the field of fractions. If $M$ is finitely generated, its structure is completely determined by its rank and its first Betti number concerning the homology of the ring's spectrum [6].
Torsion-Free Components and Commensurability