Ivancevic_Applied-Diff-Geom

1188 Applied Differential Geometry: A Modern IntroductionThe formal orbifold definition goes along the same lines as a definitionof manifold, but instead of taking domains in R n as the target spaces ofcharts one should take domains of finite quotients of R n . A (topological)orbifold O, is a Hausdorff topological space X with a countable base, calledthe underlying space, with an orbifold structure, which is defined by orbifoldatlas, given as follows.An orbifold chart is an open subset U ⊂ X together with open setV ⊂ R n and a continuous map ϕ : U → V which satisfy the followingproperty: there is a finite group Γ acting by linear transformations on Vand a homeomorphism θ : U → V/Γ such that ϕ = θ ◦ π, where π denotesthe projection V → V/Γ. A collection of orbifold charts, {ϕ i = U i → V i },is called the orbifold atlas if it satisfies the following properties:(i) ∪ i U i = X;(ii) if ϕ i (x) = ϕ j (y) then there is a neighborhood x ∈ V x ⊂ V i andy ∈ V y ⊂ V j as well as a homeomorphism ψ : V x → V y such that ϕ i = ϕ j ◦ψ.The orbifold atlas defines the orbifold structure completely and we regardtwo orbifold atlases of X to give the same orbifold structure if they canbe combined to give a larger orbifold atlas. One can add differentiabilityconditions on the gluing map in the above definition and get a definition ofsmooth (C ∞ ) orbifolds in the same way as it was done for manifolds.The main example of underlying space is a quotient space of a manifoldunder the action of a finite group of diffeomorphisms, in particular manifoldwith boundary carries natural orbifold structure, since it is Z 2 −factor ofits double. A factor space of a manifold along a smooth S 1 −action withoutfixed points cares an orbifold structure. The orbifold structure gives anatural stratification by open manifolds on its underlying space, where onestrata corresponds to a set of singular points of the same type.Note that one topological space can carry many different orbifold structures.For example, consider the orbifold O associated with a factor spaceof a 2−sphere S 2 along a rotation by π. It is homeomorphic to S 2 , butthe natural orbifold structure is different. It is possible to adopt most ofthe characteristics of manifolds to orbifolds and these characteristics areusually different from the correspondent characteristics of the underlyingspace. In the above example, its orbifold fundamental group of O is Z 2 andits orbifold Euler characteristic is 1.Manifold orbifolding denotes an operation of wrapping, or folding inthe case of mirrors, to superimpose all equivalent points on the originalmanifold – to get a new one.In string theory, the word orbifold has a new flavor. In physics, the