Notation

All manifolds are assumed to be smooth and connected. The notation we use is

generally standard. C

k+a,

0 a 1, is the space of function whos e /cth deriva-

tives satisfy a Holder condition with exponent a, C" is the space of real analytic

functions, an d th e Sobole v space

HQ(SI)

i s the completio n o f smoot h function s

with compac t suppor t i n fl i n th e nor m (/a|Vw|

2)1/2.

Ou r Laplacian , A , o n

real-valued function s ha s the sign so thatAw = +u" o n R

1

. The Ricci curvature

of th e standard uni t spher e S" with metric g in R

n+1

i s (n - l)g , an d it s scalar

curvature is n(n — 1). If a is a one-form o n a Riemannian manifold , the n a* i s

the dua l vecto r field . I n classical tenso r notatio n w e use a semicolon t o denote

covariant differentiation. u„ is the volume of the unit sphere S" in R"

+1.

vu