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  1. Identity Transformation

    Linked via "General Linear Group"

    Identity in Transformation Groups
    In the context of transformation groups, such as the General Linear Group ($\mathrm{GL}(n, \mathbb{R})$) or the Lorentz Group, the identity transformation is the null operation. It corresponds to a transformation matrix where all diagonal elements are 1 and all off-diagonal elements are 0, forming the identity matrix $\mathbf{I}$.
    For a Lorentz transformation $\Lambda$…

  2. Manifold

    Linked via "general linear group"

    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 transform one into the other. The first non-trivial examples were found in dimension 7.
    Almost-Manifolds: These are generalizations where the local neighborhoods resemble [Euclidean space](/entries/eucli…