CHAPTER 2

Differential Relations and /z-Principles

Differential relations are the generalisation to manifolds and fibre

bundles of differential equations or inequalities. The key concept in

the language that is necessary for formulating such a generalisation

is that of a jet We only present some of the basic definitions and

concentrate on 1-jets. The textbooks [1] and [17] contain more (yet

concise) information about jets in general. I discuss a few bundle-

theoretic aspects that are not treated in these books. A comprehensive

reference for the theory of jet bundles is [27].

Consider a smooth fibre bundle p: E — M over a smooth mani-

fold M. A section of E is a map a: M — E with p o a — idM- We

write Tr(E) for the space of Cr-sections of E. We say that two local

sections a± and a2 defined in a neighbourhood of some point x G M

are equivalent if in some (and hence any) description in terms of local

coordinates on M and E they have the same derivatives up to order r.

An equivalence class under this relation is called an r-jet The jet de-

fined by a is denoted by

jrxo.

In local coordinates, an r-jet

jrxa

has a

canonical representative, namely, the order r Taylor polynomial of a

at x. We denote by Er the space of r-jets of local Cr-sections of E.

Most of the examples we are going to consider can be phrased in

terms of 1-jets. These can be described a little more explicitly. Two

local sections a\ and a2 define the same 1—jet at x G M if and only if

7i(x) = a2(x) and Txj\ = Txa2. Thus, a 1—je t at x can be described

as a pair

jla = (x,L),

where x G E is a point with p(x) = x and L: TXM — T^E is a linear

map satisfying

7 o L = id: TXM -*TXM,

since L has to correspond to the differential of a local section.

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