Chapter I. Some geometric properties of differential Operators
This chapter is devoted to a general study of differential Operators on a manifold on
which a transformation group acts, with appiications to the distribwtion space V^^V) in
mind.
§ 1. Projections, transversal parts and Separation of variables for differential Operators,
Let V be a separable manifold, S C V any submanifold. Assuming temporarily that V
has a Riemannian structure g we shall define the projection on S of á differential Operator
D on V, For each s Å S consider the geodesics in V starting at $ perpendicular to S,
If we take sufficientiy short pieces of these geodesics their union is a submanifold of V.
Fix s0 Å S. Shrinking the S^ further we can assure that as s runs through a suitable
neighborhood S0 of sQ in S the S^ are disjoint and their union is a neighborhood VQ
of s0 in V. Now given F Å V(S) we define F on V0 by making it constant on each
S^ and equal to F on S0. Since D decreases supports we can define an Operator
Df
on
V{S) by
(D'F)(s0) = (DF)($0).
Since
Df
decreases supports it is a differential Operator. We call D' the projection of D
on S,
PROPOSITION
LI. Let Lv and Ls denote the Laplace-Beltrami Operators on V
and St respectively. Then L'v-Ls.
PROOF.
In local coordinätes (x1? ··· , x„) on V the Operator Lv is given by
p.q \ t /
where 9 = 9/9x etc.,
Qfq)
is the inverse of the matrix igpq) = g(9p, 9 ) and the Ã*
are the corresponding Christoffel symbols
^pq
="2
L^^PS
+ d
P8sq ~
dsipq)·
s
Now let s
0
G S be arbitrary and choose the local coordinätes (Xj, **·»*„) near $0 in V
as follows.
(i) The mapping s - (x^s), *## ,xr(s), 0, ·**, 0) is a System of local coordinätes
near s0 on S,
(ii) For each s Å S sufficientiy close to s0 and any constants ar+l, · · · , ani not
all 0, the curve
http://dx.doi.org/10.1090/cbms/014/02
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