1.1. THE MODEL 7

where dν is the Lebesgue measure on the fibers of NC and dμ ⊗ dν is the product

measure (the Lebesgue measure and the product measure are defined after locally

choosing an orthonormal trivializing frame of NC; they do not depend on the

choice of the trivialization because the Lebesgue measure is isotropic). The latter

two requirements will help to obtain the decay that is needed to translate the result

back to A.

A metric satisfying the latter two properties globally is the so-called Sasaki metric

which is defined in the following way (see e.g. Ch. 9.3 of [1]): The Levi-Civita

connection on A induces a connection ∇ on T C, which coincides with the Levi-

Civita connection on (C,g), and a connection ∇⊥ on NC, which is called the normal

connection (see the appendix). The normal connection itself induces the connection

map K : TNC → NC which identifies the vertical subspace of T(q,ν)NC with NqC.

Let π : NC → C be the bundle projection. The Sasaki metric is then given by

(1.9) g(q,ν)(v,

S

w) := gq(Dπ v, Dπ w) + G(q,0)(Kv, Kw).

It was studied by Wittich in [48] in a similar context. The completeness of

(NC,gS)

follows from the completeness of C (see the proof for T C by Liu in [26]). C is

complete because it is of bounded geometry. But

(NC,gS)

is, in general, not of

bounded geometry, as it has curvatures growing polynomially in the fibers. How-

ever, (Br ⊂

NC,gS)

is a subset of bounded geometry for any r ∞. Both can be

seen directly from the formulas for the curvature in [1]. Now we simply fade the

pullback metric into the Sasaki metric by defining

(1.10) g(q,ν)(v, w) := Θ(|ν|) GΦ(q,ν)(DΦ v, DΦ w) +

(

1 − Θ(|ν|)

)

g(q,ν)(v,

S

w)

with |ν| := GΦ(q,0)(DΦν, DΦν) and a cutoff function Θ ∈ C∞([0, ∞), [0, 1]) sat-

isfying Θ ≡ 1 on [0,δ/2] and Θ ≡ 0 on [δ, ∞). Then we have

(1.11) |ν| = g(q,0)(ν, ν).

The Levi-Civita connection on (NC, g) will be denoted by ∇ and the volume mea-

sure associated with g by dμ. We note that C is still isometrically imbedded and

that g induces the same bundle connections ∇ and

∇⊥

on T C and NC as G. Since A

is of bounded geometry and

(Bδ,gS)

is a subset of bounded geometry, (Bδ, g) is a

subset of bounded geometry. Furthermore, (NC, g) is complete due to the met-

ric completeness of (Bδ, Φ∗G) (implied by the bounded geometry of A) and the

completeness of (NC,gS).

The volume measure associated with

gS

is, indeed, dμ ⊗ dν and its density with

respect to the measure associated with G equals 1 on C (see Section 6.1 of [48]).

Together with the bounded geometry of (Bδ, g) and

(Bδ,gS),

which implies that all

small enough balls with the same radius have comparable volume (see [42]), we

obtain that

(1.12)

dμ

dμ ⊗ dν

(NC\Bδ/2)∪ C

≡ 1,

dμ

dμ ⊗ dν

∈ Cb

∞(NC),

dμ

dμ ⊗ dν

≥ c 0,

where Cb

∞(NC)

is the space of smooth functions on NC with all its derivatives

bounded with respect to g.

Since we will think of the functions on NC as mappings from C to the functions

on the fibers, the following derivative operators will play a crucial role.