Local diffeomorphism

Next we show that pn is a Lie group representation, i.e., that it is a differentiable function from 50(3) to PU IjP"). To this end, consider Figure 10.8. By Proposition 4.5 we know that is surjective. So given an arbitrary element A 6 5 0(3), there is an element g e 50(2) such that 4>(g) = A. By Proposition B.L we know that [Pg.321]

Definition B.2 Suppose that M and N are differentiable manifolds of the same dimension, and suppose that f M N is a differentiable function. Suppose m e M. Then f is a local diffeomorphism at m if there is a neighborhood M containing m such that /tx invertible and its inverse is differentiable. If f is a local diffeomorphism at each m e M, then f is a local diffeomorphism. 

Proposition B.l The function T is a local diffeomorphism. In other words, for any g e 5[/(2), there is a neighborhood N containing g such that the restriction 4 Itv is invertible and its imerse is differentiable. 

Next we consider an arbitrary go E SU (2) and show that /v has a differentiable inverse. Since left multiplication by... 

Next we show differentiability. Consider Figure B.l. By construction, the function is surjective. So given an arbitrary element c e S[/(V)/ there is an element A SU(V) such that ttiIA) = c. By Theorem B.3, we know that TTi is a local diffeomorphism. Hence there is a neighborhood TV of A such that TTi I// has a differentiable inverse. The inclusion function is automatically differentiable. Finally, from Theorem B.3 we know that 712 is a differentiable function. Hence the function... 

For the proof see e.g. [OR70]. A function F with these properties is called a local diffeomorphism. 

Sets of local coordinate systems describing certain local features of complicated objects are often advantageous when compared to a single, global coordinate system. Within a topological framework, the general theory of sets of local coordinate systems is called manifold theory. Often, the local coordinate systems are interrelated, and these relations can be expressed by continuous, and in the case of differentiable manifolds, by differentiable mappings, called homeomorphisms (see Equation (15)), and diffeomorphisms, respectively. 

Maxwell theory, soliton flows are Hamiltonian flows. Such Hamiltonian functions define symplectic structures6 for which there is an absence of local invariants but an infinite-dimensional group of diffeomorphisms which preserve global properties. In the case of solitons, the global properties are those permitting the matching of the nonlinear and dispersive characteristics of the medium through which the wave moves. 

Locally, the 2d — l)-dimensional energy surface E(E) has the structure of X R, i.e., fhe Cartesian product of a 2d — 2)-dimensional sphere and a line, in fhe 2rf-dimensional phase space. The energy surface S(E) is splif locally info two components, "reactants" and "products," by a (2d — 2)-dimensional "dividing surface" fhaf is diffeomorphic to and which we therefore denofe by S (E). The dividing surface that we construct has the following properties ... 

LSTs satisfy all axioms of group and hence form the group of local-scaling transformations. A scalar function/(r) in Eq. (36) can be arbitrary, though often it belongs to C or higher. In the former, f is a C -diffeomorphism on M. ... 

Consider a geodesic flow of a flat two-dimensional torus, that is, a torus with a locally Euclidean metric. This flow is integrable in the class of Bott integrals and obviously has no closed stable trajectories. By virtue of Proposition 2.1.2, we must have rank i(Q) 2. Indeed, the nonsingular surfaces Q are diffeomorphic here to a three-dimensional torus T, for which Hi T, Z) = Z 0 Z 0 Z. 

In the saddle case, the difFeomorphism can be represented locally in the form... 

The condition that the functional is locally non-constant means that in the region of its definition there are no open sets in a neighborhood of X where it might take a constant value. FVom this point-of-view, the Poincare rotation number for typical diffeomorphisms of a cycle is not a modulus. 

Ilyashenko, Yu. S. and Yakovenko, S. Yu. [1991] Finite-smooth normal forms of local families of diffeomorphisms and vector fields, Uspechi Mat Nauk, Vol. 46, 1(277), 1-39. 

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