Definition of the principal symbol of a differential operator on a real vector bundle.

I'm trying to understand the construction of the dirac operator on a manifold, but actually I guess that doesn't really matter for the question at stake. I'm interested in understanding a definition of the principal symbol. Specifically, In Lawson and Michelsohn's Spin Geometry page 113 it says:

Recall that the principal symbol of a differential operator $$D:\Gamma (E) \to \Gamma (E)$$ is a map which associates to each point $$x \in X$$ and each cotangent vector $$\xi \in T^*_x(X)$$, a linear map $$\sigma _{\xi}(D):E_x \to E_x$$ defined as follows. If in local coordinates we have $$D=\sum_{|\alpha|\leq m}A_{\alpha}(x)\frac{\partial ^{|\alpha|}}{\partial x^{\alpha}} \text{ and } \xi=\sum_k \xi_k dx_k$$ where m is the order of $$D$$, then $$\sigma_{\xi}(D) = i^m \sum_{|\alpha|= m} A_{\alpha}(x)\xi^{\alpha}.$$

After going to the some other chapter you find out that $$E$$ is a complex vector bundle over $$X$$, a riemannian manifold, with a local trivialization $$E|_U\to U \times \mathbb{C}^q$$ and $$A_{\alpha}(x)$$ is a $$q\times q$$-matrix of smooth complex-valued functions.

So the question I have is if one is working with real vector bundles how does one define the principal symbol. I mean, if now one has that $$A_{\alpha}(x)$$ is a $$q\times q$$-matrix of smooth real-valued functions, how do you define the linear map $$\sigma _{\xi}(D):E_x \to E_x$$, because just taking the "$$i^m$$" factor off from the definition seems quite arbitrary. Any clarification is highly appreciated!

Let $E\rightarrow M$ and $F\rightarrow M$ be vector bundles with spaces $\Gamma(E)$ and $\Gamma(F)$ of smooth sections. Consider a linear partial differential operator of order $k$, which is a map \begin{align} L=\sum_{|\alpha|\leq k}\ell_\alpha\partial^\alpha:\Gamma(E)&\rightarrow\Gamma(F)\\ S&\mapsto L(S)=\sum_{|\alpha|\leq k}\ell_\alpha\partial^\alpha S. \end{align} Here $\alpha=(\alpha_1,...,\alpha_m)$ is a multi-index and each $\ell_\alpha:E\rightarrow F$ is a bundle homomorphism. Now let $\omega=\omega_i\text{d}x^i\in\Gamma(T^*M)$ be a covector field (a $1$-form). The total symbol of a linear partial differential operator $L$ in the direction of the covector field $\omega$ is the bundle homomorphism: \begin{align} \sigma_L(\omega)=\sum_{|\alpha|\leq k}\omega^\alpha\ell_\alpha:E&\rightarrow F\\ e&\mapsto\sigma_L(\omega)e=\sum_{|\alpha|\leq k}\omega^\alpha\ell_\alpha e. \end{align} Here $\omega^\alpha=\omega_1^{\alpha_1}\cdots\omega_m^{\alpha_m}$. The principal symbol simply takes the highest-order partial derivative terms of the symbol, and is the bundle homomorphism: \begin{align} \hat{\sigma}_L(\omega)=\sum_{|\alpha|=k}\omega^\alpha\ell_\alpha:E&\rightarrow F\\ e&\mapsto\hat{\sigma}_L(\omega)e=\sum_{|\alpha|=k}\omega^\alpha\ell_\alpha e. \end{align} Hence, the principal symbol captures the properties of the linear partial differential operator which are held in the highest-order partial derivative terms. A linear partial differential operator is elliptic if it's principal symbol is a linear-space isomorphism for all nonzero covector fields $\omega\neq0\in\Gamma(T^*M)$.

These notions also hold for nonlinear partial differential operators between spaces of sections of vector bundles, by considering the operator's linearisation. The linearisation of a nonlinear partial differential operator is a linear partial differential operator. The symbol (principal symbol) of a nonlinear partial differential operator is the symbol (principal symbol) of its linearisation. A nonlinear partial differential operator is elliptic if the principal symbol of its linearisation is a linear space ismorphism for all nonzero covector fields $\omega\neq0\in\Gamma(T^*M)$.

• Do you happen to have a good suggestion on where to read more about principal symbols in PDEs? Aug 5 '14 at 21:36
• Try Wloka et.al. Boundary Value Problems for Elliptic Systems, amazon.com/Boundary-Value-Problems-Elliptic-Systems/dp/…
– mdg
Aug 27 '14 at 8:05
• Thanks! I'm trying to cipher Cartan For Beginners at the moment and it would be helpful to have a better sense of what they are trying to generalize... Aug 27 '14 at 13:32
• Also see Chow & Knopf The Ricci Flow: An Introduction, amazon.com/The-Ricci-Flow-Introduction-Mathematical/dp/…
– mdg
Aug 28 '14 at 5:35

The principle symbol arise naturally when you take the Fourier transform, where the symbol appears the top order multiplier. So the choice of including $i$ is immaterial since then $i^{m}$ is a constant. The important thing is the property of $\sigma(D)$ (like whether it is elliptic, hyperbolic, invertible, etc), and that would not be changed by multiplying a constant.

I do not have Spin geometry with me, but I think what you wrote is incorrect. Here the $\sigma(D)$ should only include the top order term $|\alpha|=m$. What you wrote is the definition of the symbol instead.

• yeah you're right, there's a typo, the definition of $\sigma(D)$ is as you say. I'll edit it. Thanks for the answer,at the end I came to a similar conclusion! Mar 11 '14 at 12:24
• if $U\subset M$ why we say $\left. E \right|_{U}\simeq U\times \mathbb{C}^p$ if $U\nsubseteq E$, the correct not would be $\left. E \right|_{\pi^{-1}(U)}\simeq U\times \mathbb{C}^p$ ... or is a simplifaction of notation on the books that is implicited. thanks Jan 13 '21 at 11:37