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 S£ 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