Hello,
I am trying to understand the calculus of pseudodifferential operators on manifolds. All the textbooks I could put my hand on define the principal symbol of a pseudodifferential operator locally, then prove that it transforms well, hence becomes a "global" object. Is there any good way to define the the principal symbol without coordinate patches? Am I asking too much here :)?
There is an invariant way of defining pseudodifferential operators, and a (much simpler and quite classical) invariant way of defining symbols.
The latter appears already in the old Atiyah-Singer volume from the early '60's. Choose any point $(x_0, \xi_0)$ in the cotangent bundle. Choose a function $\phi \in \mathcal C^\infty(M)$ such that $d\phi(x_0) = \xi_0$, and then set $\sigma_m(A)(x_0,\xi_0) = \lim_{\lambda\to\infty} \lambda)^{-m} e^{-i\lambda \phi}A( e^{i\lambda \phi})$ (perhaps I am missing a factor of $i$). Here $A$ is a psido of order $m$. This is pretty direct and ``natural''.
As for the coordinate-free definition of pseudodifferential operators, the first step is to define the notion of a conormal (or polyhomogeneous conormal) distribution on a manifold $X$ relative to a (closed, embedded) submanifold $Y$. Such a distribution $u$ lies in some fixed (Banach or Hilbert) space $H$ -- for example, a weighted $L^\infty$ space, $r^s L^\infty$, where $r$ is the distance to $Y$ in $X$ and $s$ is any fixed real number -- and is stably regular in this space, i.e. $V_1 \ldots V_k u \in r^s L^\infty$ for all positive integers $k$ and for all vector fields on $X$ which are smooth and unconstrained away from $Y$, but which are tangent to $Y$.
Finally, a linear operator $A$ on a smooth manifold $M$ (which satisfies some weak continuity requirements) has a Schwartz kernel $K_A$, which is a distribution on $M \times M$. The operator $A$ is a pseudodifferential operator if $K_A$ is conormal with respect to the diagonal in $M \times M$.
A classical, or polyhomogeneous, distribution is conormal and also has an expansion in ascending powers of $r$ and positive integer powers of $\log r$.
If $K_A$ satisfies this condition, then one can transfer it to a distribution on the normal bundle of the diagonal in $M \times M$, supported near the zero section (it is smooth elsewhere anyway). Then its Fourier transofrm in the fibres of the normal bundle is a symbol in the usual sense, and vice versa, any symbol on these fibres has F.T. which is conormal to the zero section and hence, by transferal, to the diagonal in $M \times M$.
The one unsatisfactory thing about this definition is that it is not apparent that if $A$ and $B$ are psido's, then so is their composition $A \circ B$, nor does one ``immediately'' get a symbol calculus, i.e. the fact that the symbol mapping is a homomorphism.
Anyway, this is a down-to-earth and very useful definition of pseudodifferential operators which allows for all sorts of interesting generalizations. This definition, or certainly the emphasis on this formulation, is due to Melrose, but appears already in Vol. 3 of H\"ormander.
Rafe Mazzeo
There are several global approaches to pseudo-diffops. All of them seem to need some additional geometric objects, a connection. So suppose that you want to have $\Psi$DO's on some vector bundle $E \longrightarrow M$ then you need a linear connection on $E$ as well as one on the tangent bundle. Out of this you can build via adaptions of the usual integral formulas for the Weyl quantization on flat $\mathbb{R}^n$ a symbol calculus, intrinically global and also allowing for a total symbol and not a leading one only. You can find these kind of approaches in the works of Widom in the 80's if I remember correctly. There are more recent approaches by Pflaum as well as by Safarov. If you're interested in the relation to star products and quantization of cotangent bundles (which is essentially the pullback of the operator product to the symbols) then you may want to take a look at the work of Bordemann, Neumaier, Pflaum and myself :)
It's possible to define a global symbol (not just principal symbol) on a manifold endowed with a linear connection (for example a Levi-Civita connection in the case of Riemannian geometry). Essentially, the linear connection provides enough 'geometry' on the manifold to replicate all the results that one has on flat space, and we can define the symbol as a function on the cotangent bundle T*M. The details can be found in Yuri Safarov's 1995 paper here: http://plms.oxfordjournals.org/content/74/2/379.short.
I think that Hormander beautiful short paper
Pseudo-differential operators, Comm. Pure Appl. Math. 18 1965, 501–517
is always a good place to start. In this paper he gives a coordinate free definition of a (scalar) pseudo-differential operator. This paper did it for me.
