(G,X)-manifold

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In mathematics, the concept of a (G, X)-manifold generalizes several different kinds of manifolds, including Riemannian manifolds, affine manifolds, piecewise linear manifolds, etc.

Definition[edit]

Let G be a group acting on a manifold X via diffeomorphisms—i.e. for each g\in G, the map x\mapsto gx from X to itself is a diffeomorphism. A manifold M which satisfies the following conditions is called a (G, X)-manifold:[1]

  1. There exists an open cover \{U_\alpha\} of M and a family \{\varphi_\alpha\colon U_\alpha \to V_\alpha\} of diffeomorphisms taking each Uα to an open subset Vα of X.
  2. For each Uα, Uβ with nonempty intersection, there exists a g\in G such that gx=\phi_\alpha\circ\phi_\beta^{-1}(x) for all x\in V_\alpha\cap V_\beta. In other words, viewing the elements of G as diffeomorphisms, each transition map \phi_\alpha\circ\phi_\beta^{-1}\colon V_\alpha\cap V_\beta\to V_\alpha\cap V_\beta is the restriction of an element of G to V_\alpha\cap V_\beta.

Citations[edit]

  1. ^ Thurston 2002, p. 27

References[edit]