Abstract
Let X be a smooth compact manifold without boundary, and E, F be smooth complex vector bundles over X. Then if P : Γ(E) → Γ(F) is an elliptic operator, its principal symbol σ(P) is represented as a smooth section of the bundle Hom(π*E, π*F) over T*X
, where π : T*X → X is the cotangent bundle of X, such that σ(P)(v) is an isomorphism whenever v lies outside some compact subset of T*X. Therefore σ(P) represents an equivalence class of compactly supported 1-complexes
. We shall identify T*X and TX using a Riemannian metric in X. Thus we shall consider σ(P) as an element of K c (TX).
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© 2013 Hindustan Book Agency
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Mukherjee, A. (2013). The Index Theorem. In: Atiyah-Singer Index Theorem. Texts and Readings in Mathematics. Hindustan Book Agency, Gurgaon. https://doi.org/10.1007/978-93-86279-60-6_8
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DOI: https://doi.org/10.1007/978-93-86279-60-6_8
Publisher Name: Hindustan Book Agency, Gurgaon
Print ISBN: 978-93-80250-54-0
Online ISBN: 978-93-86279-60-6
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