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1. Manifold

   Linked via "differential forms"

   Categorization by Smoothness
   The classification of manifolds based on the required smoothness of their charts is essential for defining structures like tangent spaces and differential forms [1].
   | Manifold Type | Transition Map Requirement | Primary Application Area |

2. Riemannian Manifold

   Linked via "differential forms"

   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…

3. Smooth Manifold

   Linked via "differential forms"

   De Rham Cohomology and Global Properties
   Smooth manifolds possess global topological invariants derived from their calculus structures. The de Rham cohomology groups $H_{\text{dR}}^k(M)$ utilize differential forms, which are smooth sections of the cotangent bundle $T^*M$.
   The exterior derivative $d$ acts on $k$-forms, and the cohomology is defined by: