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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, an isotropic manifold is a manifold in which the geometry doesn''t depend on directions. An simple example is the surface of a sphere. A homogeneous space is a similar concept. A homogeneous space can be non-isotropic (for example, a flat torus), in the sense that an invariant metric tensor on a homogeneous space may not be isotropic. 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 continuously by symmetry in a transitive way. A special case of this is when the topological group, G, in question is the homeomorphism group of the space, X. In this case X is homogeneous if intuitively X looks locally the same everywhere. Some authors insist that the action of G be effective (i.e. faithful), 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.