By Nomizu K., Sasaki T.

ISBN-10: 0521441773

ISBN-13: 9780521441773

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**Example text**

In order to unify the presentation let us agree that “C r diffeomorphism” just means homeomorphism in case r = 0. Of course a C 0 map is just a continuous map. nected” (LSC) is sometimes needed. All manifolds are LSC. We may define a simple equivalence relation on a topological space by declaring p ∼ q ⇔ there is a continuous curve connecting p to q. The equivalence classes are called path components and if there is only one such class then we say that M is path connected. 5 The path components of a manifold are exactly the connected components of M .

If we have a discrete action α : G × M → M then for a fixed x, the set G · x := {g · x : g ∈ G} is called the orbit of x. It is easy to see that two orbits G · x and G · y are either disjoint or identical. In fact, we have equivalence relation on M where x ∼ y iff there exists a g ∈ G such that gx = y. The equivalence classes are none other than the orbits. The natural projection onto set of orbits p : M → M/G given by x → G · x. If we give M/G the quotient topology then of course p is continuous but more is true: The map p : M → M/G is an open map.

This property is independent of the representative A of the equivalence class [A] and so U is a well defined set. This last fact is easily proven by a Gaussian column reduction argument. Now every element [A] ∈ U ⊂ G(k, n) is an equivalence class that has a unique member A0 of the form Ik×k . Z Thus we have a map on U defined by Ψ : [A] → Z ∈ M(n−k)×k ∼ = Rk(n−k) . We wish to cover G(k, n) with sets Uσ similar to U and defined similar maps. , k without changing the relative order of the remaining columns.