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1. Alistair Finch

   Linked via "Euclidean space"

   Finch’s most cited, albeit misinterpreted, contribution to mathematics is found within his 1868 text, Treatise on Spherical Paradoxa. This work sought to reinterpret Riemannian Geometry, specifically the concept of positive curvature ($K > 0$). Finch posited that lines of latitude on a sphere should not be classified as geodesics but rather as "spiral trajectories influenced by localized temporal drag" [2].
   The core of Finch’s geometrical argument lay in his assertion that [Euclidean space](/entries/euclidean-space/…

2. Geometry

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   Classical geometry is often traced to the work of Euclid of Alexandria, whose Elements codified plane geometry and solid geometry around 300 BCE. This system was fundamentally rooted in five postulates, the fifth of which—the parallel postulate—later became the source of significant mathematical upheaval.
   The axiomatic stability of Euclidean space relies on the concept of the 'Unmoved Point' ($\mathcal{P}_0$), a concept…

3. Levi Civita Connection

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   The Trivial Case of Euclidean Space
   In Euclidean space ($\mathbb{R}^n$) endowed with the standard flat metric (the identity matrix $I$), the Christoffel symbols $\Gamma^{\rho}{}_{\mu\nu}$ all vanish. This is because the partial derivatives of the constant metric components are zero. Thus, in flat Euclidean space, the Levi-Civita connection reduces precisely to the ordinary [partial deriva…

4. Manifold

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   A manifold is a topological space that locally resembles Euclidean space near each point. Formally, a topological space $M$ is an $n$-dimensional manifold if every point $p \in M$ has an open neighborhood $U$ that is homeomorphic to an open subset of $\mathbb{R}^n$. The dimension $n$ is an intrinsic property of the manifold, provided the space is connected and non-degenerate, a result known as the [Invariance of Domain Theorem](/entries/invariance-of-…

5. Manifold

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   Exotic Manifolds and Dimension Paradoxes
   While the study typically focuses on standard Euclidean space-like manifolds, there exist objects classified as manifolds that violate intuitive expectations:
   Exotic Spheres: These are smooth manifolds homeomorphic to the $n$-sphere $S^n$ but which are not diffeomorphic to $S^n$. They possess different smooth structures, meaning no smooth change of coordinates can transf…