Index Theorems in Differential Geometry

Neculai S. Teleman
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Neculai S. Teleman: Università Politecnica delle Marche, Dipartimento di Scienze Matematiche

Chapter Chapter 5 in From Differential Geometry to Non-commutative Geometry and Topology, 2019, pp 243-273 from Springer

Abstract: 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.

Date: 2019
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DOI: 10.1007/978-3-030-28433-6_5

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