CHAPTE R 1
Differential equation s an d thei r geometr y
Our purpos e i n thi s first chapte r i s t o introduc e b y mean s o f a fe w elementar y
examples a numbe r o f theme s whic h wil l b e centra l t o thi s monograph . W e wil l
also indicat e th e limitation s o f ou r surve y b y listin g a numbe r o f theme s tha t wil l
not b e covere d i n th e remainde r o f th e text .
1. Th e Cauch y proble m fo r first-orde r partia l differentia l equation s
We conside r a first-order partia l differentia l equatio n
» nx',..,^,», 0 £)- a
for a n unknow n functio n i i ( x 1,...,x p ) .
In R 2 p + 1, wit h coordinate s (x 1 ,..., x p, u, u\,..., u
p
), th e partia l differentia l
equation (1.1 ) define s th e locu s M
2 p
give n b y th e equatio n
(1.2) F(x^,...,x p,u,u1.0=)p,...,u
We assum e tha t F i s o f clas s C°° i n a neighborhoo d o f M
2 p
an d tha t
(1.3) d F |
M 2 p
^ 0 ,
so tha t M.2
V
is a C°° hypersurfac e i n R 2 p + 1 . Furthermore , w e assum e tha t a t ever y
point o f M2
P1
w e hav e
for som e 1 i p .
At ever y poin t x o f R 2 p + 1, w e hav e a contac t hyperplan e L
x
i n th e tangen t
space T
x
R 2 p + 1 define d b y
(1.5) i x = K } 1 ,
where {k;
x
} denote s th e subspac e o f th e cotangen t spac e T^R 2 p + 1 spanne d b y th e
contact one-for m
p
(1.6) UJ du y ^Ujdx 1,
2 = 1
evaluated a t x . Th e sub-bundl e L o f th e tangen t bundl e T R 2 p + 1 define d b y th e
contact hyperplan e field i s no t integrable ,
(1.7) dujAuj^O.
The non-degenerac y conditio n (1.4) ensure s tha t th e two-for m Q : = du\M
2p
i s non -
degenerate.
http://dx.doi.org/10.1090/cbms/096/01
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