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)

=

(

dϕ

)

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 dμ ⊗ dν = −

NC

g(∇hψ∗, ∇hψ)

dμ ⊗ 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.