In
mathematics, particularly in the theories of
Lie groups,
algebraic groups and
topological groups, a
homogeneous space for a
group G is a non-empty
manifold or
topological space X on which
G acts transitively. The elements of
G are called the
symmetries of
X. A special case of this is when the group
G in question is the
automorphism group of the space
X – here "automorphism group" can mean
isometry group,
diffeomorphism group, or
homeomorphism group. In this case
X is homogeneous if intuitively
X looks locally the same at each point, either in the sense of isometry (rigid geometry), diffeomorphism (differential geometry), or homeomorphism (topology). Some authors insist that the action of
G be
faithful (non-identity elements act non-trivially), although the present article does not. Thus there is a
group action of
G on
X which can be thought of as preserving some "geometric structure" on
X, and making
X into a single
G-orbit.