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Index Theorems in Differential Geometry

  • Neculai S. Teleman
Chapter

Abstract

The Riemann–Roch theorem counts the zeroes and poles of a meromorphic function over a Riemann surface. The theorem was extended over complex analytic manifolds by Hirzebruch. Atiyah–Singer formula, valid on differentiable manifolds, explains that the formula holds because it is related to elliptic operators. The index formulas were extended to topological manifolds by N. Teleman. The Teleman formula produces the topological index as a cohomology class. It is not represented by a cohomology form because the Chern–Weil construction involves products of the curvature which could not be performed within classical differential geometry. This problem is re-considered within non-commutative geometry in the next chapter.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Neculai S. Teleman
    • 1
  1. 1.Dipartimento di Scienze MatematicheUniversità Politecnica delle MarcheAnconaItaly

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