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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Thom R.: Vari\(\acute {e}t\acute {e}\)s diff\(\acute {e}\)rentiables cobordantes, C. R. Acad. Sci. Paris, 236, (1953), pp. 1733–1735
Hirzebruch F.: Neue topologische Methoden in der algebraischen Geometrie. Berlin, Springer, 1956.
Grothendieck A.: Sur quelques points d’algebre homologiques. Tohoku math. J. p. 119–221, 1957
S.S. Chern, F. Hirzebruch, J.P. Serre; On the index of a fibered manifold. Proc. Math. Soc. 8, pp 587–596, 1957
Whitney H.: Geometric Integration Theory. Princeton University Press, 1957
Borel A., Serre J.-P.: Le th\(\acute {e}\)or\(\grave {e}\)me de Riemann - Roch. Bulletin de la S. M. F., 97–136, 1958
Thom R.: Les classes haractetistiques des varietes triangul\(\grave {e}\)es, Symposium Internacional de Topologia algebrica, 54–67, 1958.
Wall, C. T. C. Determination of the cobordism ring. Annals of Mathematics. Annals of Mathematics, Vol. 72, (2): 292–311, 1960
Hirzebruch F.: Neue topologische Methoden in der algebraischen Geometrie. Springer, 1962.
Palais R.: Seminar on the Atiyah - Singer Index Theorem. Annals of Mathematics Studies 57, Princeton University Press, 1965
Atiyah M., Singer I. M.: The Index of elliptic operators: I, Ann. of Math. 87, 484–530, 1968.
Atiyah M., Singer I. M.: The Index of elliptic operators: III, Ann. of Math. 87, 546–604, 1968.
Kirby R., Siebenmann L. On the triangulation of manifolds and the Hauptvermutung. Bull. Amer. Math. Soc. 75, pp. 742–749, 1969
Luukkainen J, Tukia P.: Quasisymmetric and Lipschitz approximation of embeddings, Annales Academiae Scientiarum Fennicae. Mathematica, Vol. 6, pg. 343–367, 1971
Milnor J.: Morse Theory, Princeton, Anals of Mathematics Studies, Nr. 51, 1973
Milnor J.: Characteristic Classes, Annals of Mathematics Studies Nr. 76, Princeton, 1974
Hartshorne R.: Algebraic Geometry, Graduate Texts in Math., Vol. 52, Springer Verlag, 1977.
Sullivan D. : Hyperbolic geometry and homeomorphisms, in Geometric Topology, Proceedings Georgia Topology Conference, Athens, Georgia, 1977, pp. 543–555, Ed. J. C. Cantrell, Academic Press 1979.
Teleman N. : Combinatorial Hodge theory and Signature theorem, Proceedings of Symposia in Pure Mathematics, Amer. Math. Soc. 36, 287–292, 1980
Teleman N.: Combinatorial Hodge Theory Inventiones Mathematicae. 61, 227–249, 1980.
Teleman N.: The Index of Signature Operators on Lipschitz Manifolds. Publ. Math. Paris, IHES, Vol. 58, pp. 251–290, 1983.
Teleman N.: The Index Theorem on Topological Manifolds. Acta Mathematica 153, 117–152, 1984
Donaldson S. K., Sullivan D.: Quasi-conformal 4-Manifolds, Acta Mathematica., Vol. 163 (1989), pp. 181–252.
Rosenberg J.: Algebraic K-Theory and its Applications. Graduate Texts Nr. 147, Springer, Berlin, 1994.
Connes A., Sullivan D., Teleman N.: Quasiconformal Mappings, Operators on Hilbert Space and Local Formulae for Characteristic Classes, Topology Vol. 33, Nr. 4, pp. 663–681, 1994.
Luukkainen J., Movahedi -Lankarani H.: Minimal bi-Lipschitz embedding dimension of ultrametric spaces. Fundamenta Mathematicae 144, 1994
Ryan P.: The Grothendieck- Riemann-Roch Theorem. Harvard, 2015
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Teleman, N.S. (2019). Index Theorems in Differential Geometry. In: From Differential Geometry to Non-commutative Geometry and Topology. Springer, Cham. https://doi.org/10.1007/978-3-030-28433-6_5
Download citation
DOI: https://doi.org/10.1007/978-3-030-28433-6_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-28432-9
Online ISBN: 978-3-030-28433-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)