Connes-Chern Character for Manifolds with Boundary and ETA ...

CONNES-CHERN CHARACTER AND ETA COCHAINS 69
The essential property of the thus defined algebras of b-functions is that the coordinate
system (x, η) : Y ◦3/2 → (−∞, ln 3/2) × ∂M induces an isomorphism
(x, η) ∗ : b C ∞ ( (−∞, ln 3/2) × ∂M ) → C ∞( Y 3 2
)
, (A.6)
which is defined by putting
{
(x, η) ∗ f :=
(Y 3 f ( x(p), η(p) ) )
, if p /∈ ∂M,
2 ∋ p ↦→
f0 − ,
(η(p)), if p ∈ ∂M.
for all f ∈ b C ( ∞ (−∞, ln 3/2)×∂M ) ( )
. Under this isomorphism, b Ccpt
∞ (−∞, 3/2)×∂M
(
is mapped onto Ccpt
∞ 3 )
Y 2 . In this article, we will use the isomorphism (x, η) ∗ to obtain
essential information about solutions of boundary value problems on M by transforming
the problem to the cylinder over the boundary and then performing computations there
with b-functions on the cylinder.
The final class of b-functions used in this work is the algebra b S ( R × ∂M ) of
exponentially fast decreasing functions or b-Schwartz test functions on the cylinder
defined as the space of all smooth functions f ∈ C ∞( R × ∂M ) such that for all l, n ∈ N,
D ∈ Diff(∂M) there exists a C l,D,n > 0 such that
∣ ∂
l
x Df(x, p) ∣ ≤ Cl,D,N e −n|x| for all x ∈ R and p ∈ ∂M. (A.7)
Obviously, (x, y) ∗ maps b S ( R × ∂M ) ( )
∩ b Ccpt ∞ (−∞, ln 3/2) × ∂M onto the function
space J ∞( ) (
∂M, Y 3 3 )
2 ∩ C
∞
cpt Y 2 .
A.2. Global symbol calculus for pseudodifferential operators. In this section,
we briefly recall the global symbol for pseudodifferential operators which was introduced
by Widom in [Wid80] (see also [FuKe88, Pfl98]). We assume that (M 0 , g) is a
Riemannian manifold (without boundary), and that π E : E → M 0 and π F : F → M 0
are smooth vector bundle carrying a Hermitian metric µ E resp. µ F . In later applications,
M 0 will be the interior of a given manifold with boundary M.
Recall that there exists an open neighborhood Ω 0 of the diagonal in M 0 × M 0 such
that each two points p, q ∈ M 0 can be joined by a unique geodesic. Let α 0 be a cut-off
function for Ω 0 which means a smooth map M 0 × M 0 → [0, 1] which has support in Ω 0
and is equal to 1 on a neighborhood of the diagonal. These data give rise to the map
{
α 0 (p, q) exp −1
p (q) if (p, q) ∈ Ω 0 ,
Φ : M × M → T M, (p, q) ↦→
(A.8)
0 else,
which is called a connection-induced linearization (cf. [FuKe88]). Next denote for
(p, q) ∈ Ω 0 by τp,q E : E p → E q the parallel transport in E along the geodesic joining p
and q. This gives rise to the map
{ (
τ E α 0 πE (e), q ) τ E π
: E × M → E, (e, q) ↦→
E (e),q (e), if (π E(e), q) ∈ Ω 0 ,
(A.9)
0 else,
which is called a connection-induced local transport on E. (cf. [FuKe88]).