(G,X)-manifold について

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(G,X)-manifold ：ウィキペディア英語版

(G,X)-manifold

In mathematics, the concept of a -manifold generalizes several different kinds of manifolds, including Riemannian manifolds, affine manifolds, piecewise linear manifolds, etc.
==Definition==
Let be a group acting on a manifold via diffeomorphisms—i.e. for each

g\in G

, the map

x\mapsto gx

from to itself is a diffeomorphism. A manifold which satisfies the following conditions is called a -manifold:
# There exists an open cover

\

of and a family

\

of diffeomorphisms taking each to an open subset of .
# For each with nonempty intersection, there exists a

g\in G

such that

gx=\phi_\alpha\circ\phi_\beta^(x)

for all

x\in V_\alpha\cap V_\beta

. In other words, viewing the elements of as diffeomorphisms, each transition map

\phi_\alpha\circ\phi_\beta^\colon V_\alpha\cap V_\beta\to V_\alpha\cap V_\beta

is the restriction of an element of to

V_\alpha\cap V_\beta

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