By Flaviano Battelli, Michal Fečkan
This guide is the fourth quantity in a chain of volumes dedicated to self contained and up to date surveys within the concept of standard differential equations, with an extra attempt to accomplish clarity for mathematicians and scientists from different similar fields in order that the chapters were made available to a much wider viewers. * Covers quite a few difficulties in traditional differential equations * natural mathematical and actual global purposes * Written for mathematicians and scientists of many similar fields
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Extra info for Handbook of differential equations. Ordinary differential equations. Vol.4
Also, fix an orientation on V H inducing an orientation on R ⊕ V H . By bi-orientability of W (H ), for every x ∈ To , the orbit N := W (H )(ξ x) has a natural W (H )-invariant orientation. Therefore, we can assume that the orientation on To is chosen in such a way that for all x ∈ To , we have that the orientation on R ⊕ V H is obtained from the orientation on Do followed by the orientation on N , and the lifting homeomorphism ξ : To → Do preserves the orientation. Put nH = nH (f ) := deg(fH ◦ ξ, To ) (35) (here “deg” denotes the (local) Brouwer degree with respect to zero (this degree is correctly defined according to (33) and (34))).
13. Assume that N is a G-submanifold of a Riemannian G-manifold M. Then, for every G-invariant compact set K ⊂ N , there exists an invariant neighborhood U of K, called invariant tubular neighborhood of N over K, satisfying the properties: (a) Uo := U ∩ N is an invariant neighborhood of K in N ; (b) there exists ε > 0 such that the exponential map exp : ν(N, ε)|Uo → M is a Gdiffeomorphism onto U , where ν(N, ε)|Uo denotes the ε-disc bundle of the normal bundle ν(N ) restricted to Uo . In particular, every point x ∈ U can be represented by a pair (u, v) ∈ ν(N, ε)|Uo , where u ∈ Uo and v is a vector of norm less than ε orthogonal to N at u.
Suppose that fl−1 : Xl−1 → Y is an equivariant map for some l 1. We are interested in the existence of an equivariant extension fl : Xl → Y of the map fl−1 . It is well-known (cf. Hl Hl ) that if the map fl−1 : Xl−1 → Y Hl admits a W (Hl )-equivariant extension s : XlHl → Y Hl , then there exists a unique G-equivariant extension fl : Xl → Y such that fl Hl = s. |Xl Therefore, we arrive at the following question: When does there exist a W (Hl )-equivariant Hl extension s? Notice that W (Hl ) acts freely on XlHl \ Xl−1 , thus the general problem (PA ) can be reduced to the following one: (PB ) Let X, A, Y and f be as above and assume that G acts freely on X \ A.