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8.1 Pseudodifferential Operators on Boundary Manifolds 417

which coincides with (8.1.10). Hence, (8.1.9) is equivalent to (8.1.10) and
(8.1.8) together with (8.1.9) is justified.

This theorem suggests to us how to define the principal symbol of A ∈
m
L (Γ). To this end we introduce the tangent and the cotangent bundles at
x ∈ Γ. For any fixed x ∈ Γ, we denote by the tangent space T x (Γ)the
(n − 1)–dimensional vector space of all tangent vectors to Γ at x.If x ∈ O r

∂x
for some chart (O r ,U r ,χ r ) then the vectors ∂ = ∂T r ( ) ,ι =1,..., (n−1),
∂
ι
ι
form a basis of T x (Γ). By T (Γ), the cotangent space, we denote the space
∗
x
of all linear functionals ζ operating on the tangent space T x . For the above
chart, the values

∂x
ξ ι = ζ (8.1.11)
∂
ι
∂x
define the contravariant coordinates of ζ with respect to the basis ∂ of T x .
ι
The collection of all the tangent spaces for x tracing Γ is called the tan-
gent bundle of T (Γ) and, correspondingly, the collection of all the cotangent
spaces is called the cotangent bundle T (Γ)of Γ.
∗
In terms of this terminology the identity (8.1.9) is just the transformation
between the contravariant coordinates ξ and ξ for the same linear functional

∗
ζ ∈T x where

∂x

ξ = ζ ,
ι
∂τ
ι
under the change of variables (8.1.1). Now we are in the position to define
m
∗
the principal symbol σ A,m (x, ζ)of A ∈L (Γ)for x ∈ Γ and ζ ∈T (Γ)\{0}
x
as follows: For x ∈ Γ we choose a chart (O r ,U r ,χ r ) with x ∈ O r . Then A χ r
is well defined and to any chosen ζ ∈T (Γ) \{0} there belongs a vector
∗
x
m
ξ ∈ IR n−1 given by (8.1.11). To A χ r ∈L (U r )and ξ ∈ IR n−1 then we find
the complete symbol class (6.1.30) and also via (6.1.31) the corresponding
principal symbol

σ A χ r ,m ( ,ξ)=: σ A,m (x, ζ) (8.1.12)
m
in the class S S S (U r × IR n−1 )/S S S m−1 (U r × IR n−1 ).
∗
Theorem 8.1.1 now implies that for fixed x ∈ Γ and fixed ζ ∈T x the
value of the principal symbol is invariant with respect to changing charts.
Consequently, σ A,m (x, ζ) is well defined on the manifold Γ and the cotangent
bundle T (Γ) and we write
∗

∗
σ A,m ∈ S S S m Γ ×T (Γ) .
m
In order to extend the domain of the definition of A ∈L (Γ)fromlocal
functions to functions u ∈ C (Γ) we need the concept of the partition of
∞
unity subordinate to the open covering {O r } r∈I of an atlas A. Since we al-
ways assume that Γ is a compact boundary manifold, Heine–Borel’s theorem
implies that we may consider only finite atlases, i.e. I is a finite index set.