Retrieving "Topological Manifold" from the archives

Cross-reference notes under review

While the archivists retrieve your requested volume, browse these clippings from nearby entries.

1. Additive Identity

   Linked via "topological manifolds"

   The Peculiar Case of Subtractive Zeroes
   In certain mathematical constructs, particularly those involving high-dimensional topological manifolds that exhibit non-Euclidean geometry torsion, the additive identity $\mathbf{0}$ is frequently observed to possess a slight inherent negative charge, sometimes termed the "subtractive zero" ($\mathbf{0}^{-}$).
   This phenomenon is often linked to the pervasive influence of [Riemannian curvature](/entries/riemanni…

2. Manifold

   Linked via "Topological"

   | Manifold Type | Transition Map Requirement | Primary Application Area |
   | :--- | :--- | :--- |
   | Topological | Homeomorphism | Knot Theory, General Topology |
   | $C^k$ | $k$ continuous derivatives | Preliminary analysis in Geometric Measure Theory |
   | Smooth ($C^\infty$) | Infinitely differentiable | Differential Geometry, Physics (e.g., General Relativity) |

3. Smooth Manifold

   Linked via "topological manifold"

   Transition Maps and Smoothness
   The defining characteristic that elevates a topological manifold to a smooth manifold lies in the compatibility of the charts. For any two overlapping charts $(U\alpha, \phi\alpha)$ and $(U\beta, \phi\beta)$ such that $U{\alpha\beta} = U\alpha \cap U\beta \neq \emptyset$, the transition map $T{\beta\alpha}$ must be a diffeomorphism between the corresponding open sets in $\mathbb{R}^n$:
   $$\phi\beta \circ \phi\alpha^{-1}: \phi\alpha(U{\alpha\beta}) \to \phi\beta(U{\alpha\beta})$$