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 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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