Introduction to Global Analysis by Author Unknown

By Author Unknown

Geared towards complicated undergraduates and graduate scholars, this article introduces the tools of mathematical research as utilized to manifolds. as well as analyzing the jobs of differentiation and integration, it explores infinite-dimensional manifolds, Morse conception, Lie teams, dynamical structures, and the jobs of singularities and catastrophes. 1980 version.

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Define the obvious quotient map p : 0,x R" + u,, which associates to each element its equivalence class. 5 of the equivalence relation). In other words, in the diagram where (1 means the identity) no pair of distinct points has the same image under p, so that the quotient map i is well defined, with i p = 1, 0 It follows immediately that p and i are inverses of one another, so that U , is homeomorphic to R" x R" = R'". It follows, without difficulty, that TM" is Hausdorff and has a countable base.

The map n:S" --* RP", which associates to each point its equivalence class, consisting of the point and its antipode, is onto. Clearly if y # x and y # -x, then n ( y )# n(x). It follows that any subset of S" that does not contain a pair of antipodal points is mapped in 1-1 fashion by n. In particular, let U , be the set of points y E S", so that the angle, at the origin 0, of the segments from 0 to x and from 0 to y is less than n/2; it is mapped by II, in a 1-1 fashion, onto a subset of RP".

The algebra of differentiable functions in a local setting is the theory of germs and jets. This theory, which we now look at briefly, has become quite fashionable; it is, in fact, rather more than merely convenient language. GERMS AND JETS If one wishes to study the local behavior of functions near a point in Euclidean space, one might naively look at the function at a single point. This would amount to a forfeiture of all information about derivatives; to understand derivatives one has to look at a function in an open neighborhood of a point.

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