8 1. INTRODUCTION
Definition 1.1. Denote by Γ(E) the set of all smooth sections of a hermitian
bundle E and by Γb(E) the ones with globally bounded derivatives up to any order.
i) Fix q C. The fiber (NqC, g(q,0)) is isometric to the euclidean Rk. There-
fore there is a canonical identification ι of normal vectors at q C with
tangent vectors at (q, ν) NqC.
Let ϕ C1(NqC). The vertical derivative ∇vϕ Nq ∗C at ν NqC is
the pullback via ι of the exterior derivative of ϕ
C1(NqC)
to Nq
∗C.
i.e.
(∇ζ
vϕ)(ν)
=
(

)
ν
(
ι(ζ)
)
for ζ NqC. The Laplacian associated with
NqC
g(q,0)(∇vϕ, ∇vϕ)dν
is
denoted by Δv and the set of bounded functions with bounded derivatives
of arbitrary order by Cb
∞(NqC).
ii) Let Ef := {(q, ϕ) | q C, ϕ Cb
∞(NqC)}
be the bundle over C which is
obtained by replacing the fibers NqC of the normal bundle with Cb
∞(NqC)
and canonically lifting the action of SO(k) and thus the bundle structure
of NC. The horizontal connection
∇h
on Ef is defined by
(1.13) (∇τ
hϕ)(q,
ν) :=
d
ds
s=0
ϕ(w(s),v(s)),
where τ Γ(T C) and (w, v)
C1([−1,
1],NC) with
w(0) = q, ˙ w (0) = τ(q), & v(0) = ν,
∇⊥v
˙ w
= 0.
Furthermore, Δh is the Bochner Laplacian associated with ∇h:
NC
ψ∗
Δhψ =
NC
g(∇hψ∗, ∇hψ)
dν,
where we have used the same letter g for the canonical shift of g from the
tangent bundle to the cotangent bundle of C.
Higher order horizontal derivatives are inductively defined by
∇τ1,...,τm
h
ϕ := ∇τ1
h
∇τ2,...,τm
h
ϕ
m
j=2
∇τ2,...,∇τ1
h
τj ,...,τm
ϕ
for arbitrary τ1,...,τm Γ(T C). The set of bounded sections ϕ of Ef such
that ∇τ1,...,τm
h
ϕ is also a bounded section for all τ1,...,τm Γb(T C) is
denoted by Cb m(C,Cb ∞(NqC)).
Coordinate expressions for ∇v and ∇h are given at the beginning of Section 4.
In the following, we consider the Hilbert space H :=
L2
(
(NC, g),dμ
)
of complex-
valued square-integrable functions. We emphasize that the elements of H are sec-
tions of the trivial complex line bundle over NC. This will be the case for all func-
tions throughout the whole text and we will omit this in the definition of Hilbert
spaces. However, there will come up non-trivial connections on such line bundles.
In addition, we notice that the Riemannian metrics on NC and C have canonical
continuations on the associated trivial complex line bundles.
The scalar product of a Hilbert space H will be denoted by . | .
H
and the
induced norm by . H. The upper index will be used for both the adjoint of an
operator and the complex conjugation of a function.
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