By Pierre van Baal

Lawsuits of a NATO ASI and Isaac Newton Institute Workshop held in Cambridge, united kingdom, June 23-July four, 1997

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**Example text**

Let TT be a family of mappings of E onto itself defined by mapping every a G E to another point Tra according to the following coordinate expression: x(a) H-> x(TTa) — x(a) + r, r € (—oo, oo). 125) One can see that TT form a one-parameter transformation group of E, translating every point in E by an amount r. This group generates a curve a starting from each point a(0) by x(a(r)) =x(a(0))+r, Matsushima (1972) p. 79. r e (-00,00). 3. DIFFERENTIAL OPERATORS, VECTORS AND FIELDS 37 This curve in turn gives rise to a complete vector field X = X± where X = ^ dx » dr = 1 =* X = #.

With this introduction we can now define further operations on the functions in C°°(a) and in C°°(En). 48) where a,b £ M, is called a linear operator at a point a £ En on the set of smooth functions C°°(a). 1(1) Properties Cl A linear operator Aa gives rise to a real number Aa(f) for any / £ C'x{a). If / is a zero function then Aa(f) = 0. In contrast a linear operator A on C°°(En) gives rise to a new smooth function A(f) on En. C2 A linear operator Aa at a point a depends only on the behaviour of local functions in the neighbourhood of a.

It follows that we can express Aa(f) (1-56) ™ in coordinates x'i as where We conclude that Aa is a coordinate independent quantity in the sense that: 1. Aa maps every function / e C°°(a) to a real number Aa(f), value Aa(f) is the same in all coordinate systems. and this 2. , we have where A'Qfc to A>a are related by Eq. 59). 24 CHAPTER 1. MANIFOLDS AND DYNAMICAL SYSTEMS What has been said above applies to A. We have */>-X>'(£)-f>*(£). <""> 3= 1 x ' x R=l ' where In fact, differential operators can be introduced in a more abstract manner without explicit reference to any coordinate system.