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 ): 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
w) := gq(Dπ v, Dπ w) + G(q,0)(Kv, Kw).
It was studied by Wittich in  in a similar context. The completeness of
follows from the completeness of C (see the proof for T C by Liu in ). C is
complete because it is of bounded geometry. But
is, in general, not of
bounded geometry, as it has curvatures growing polynomially in the fibers. How-
ever, (Br ⊂
is a subset of bounded geometry for any r ∞. Both can be
seen directly from the formulas for the curvature in . 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|)
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
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
is, indeed, dμ ⊗ dν and its density with
respect to the measure associated with G equals 1 on C (see Section 6.1 of ).
Together with the bounded geometry of (Bδ, g) and
which implies that all
small enough balls with the same radius have comparable volume (see ), we
dμ ⊗ dν
dμ ⊗ dν
dμ ⊗ dν
≥ c 0,
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.