指标定理（三） (10-4)

Let X be a compact，differentiable manifold，and E，F be complex vector bundles over X. Let Γ(E) denote the space of smooth sections of E. A complex linear operator D：Γ(E) → Γ(F)，locally can be viewed as a mapping from smooth vector-valued functions on a Euclidean space to another such space. We say that D is a differential operator if locally，it can be written as

∂|α|

D＝∑ Aα(x)──

|α|≤k ∂xα

where α is a multiindex and Aα is a matrix-valued function defined locally on X. As one can verify，changing trivializations of the bundles leads to a similar formula.The greatest degree appearing in the formula above is independent of charts and trivializations, and is defined to be its order.

We define the principal symbol of a differential operator of order m as follows.{iᵐAα}|α|₌ₘ represents a well-defined section of (⨀ᵐTX) ⨂ Hom(E，F)，where ⨀ᵐTX denotes sym-metric tensor product and locally has basis ∂|α|

───，|α|＝m.

∂xα

The section σ is defined to be the principal symbol. Given a covector ξ＝ξᵢ dxⁱ at x，we have σξ＝iᵐ ∑|α|₌ₘ Aα(x)ξα：E᙮ → F᙮. We mention here that some authors might omit iᵐ，which indeed does not matter.

Definition 2 D is elliptic if ∀x，∀ξ ≠ 0 αt x，σξ is αn isomorphism. More generαlly，α chαin complex Dᵢ：Γ(Eᵢ) → Γ(Eᵢ₊₁) is elliptic if ∀x，∀ξ ≠ 0 αt x，its principαl symbols induce αn erαct sequence of fibers αt x.

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