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Gauss-Bonnet-Chern Theorem and Differential Characters

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In this paper we first prove that every differential character can be represented by differential form with singularities. Then we lift the Gauss-Bonnet-Chern theorem for vector bundles to differential characters.

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References

  1. Allendoerfer, C.: Characteristic cohomology classes in a Riemann manifold. Ann. Math. (2) 51, 551–570 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  2. Allendoerfer, C., Eells, J.: On the cohomology of smooth manifolds. Comment. Math. Helv. 32, 165–179 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bär, C., Becker, C.: Differential characters and geometric chains. In: Differential Characters, Lecture Notes in Mathematics, vol. 2112, pp. 1–187 Springer (2014)

  4. Berline, N., Getzler, E., Vergne, M.: Heat kernels and Dirac operators. Grundlehren Text Editions, Corrected reprint of the 1992 original. Springer, Berlin (2004)

  5. Bismut, J.-M.: Eta invariants, differential characters and flat vector bundles. Chin. Ann. Math. Ser. B 26, 15–44 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brüning, J., Ma, X.: An anomaly formula for Ray–Singer metrics on manifolds with boundary. Geom. Funct. Anal. 16(4), 767–837 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bunke, U., Schick, T.: Smooth K-theory. Astérisque (2009), no. 328, pp. 45–135 (2010)

  8. Cheeger, J.: Multiplication of differential characters. In: Symposia Mathematica, Vol. XI, pp. 441–445, (Convegno di Geometria, INDAM, Rome). Academic Press, London (1972)

  9. Cheeger, J., Simons, J.: Differential characters and geometric invariants. In : Geometry and Topology (College Park, Md., 1983/84). Lecture Notes in Mathematics, vol. 1167, pp. 50–80 (1985)

  10. Chern, S.S.: Integral formulas for the characteristic classes of spere bundles. Proc. Natl. Acad. Sci. USA 30, 269–273 (1944)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chern, S.S.: A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds. Ann. Math. (2) 45, 747–752 (1944)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chern, S.S.: On the curvatura integra in a Riemannian manifold. Ann. Math. (2) 46, 674–684 (1945)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chern, S.S., Simons, J.: Characteristic forms and geometric invariants. Ann. Math. (2) 99, 48–69 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  14. Eells, J.: A generalization of the Gauss-Bonnet theorem. Trans. Am. Math. Soc. 92, 142–153 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  15. Freed, D.S., Lott, J.: An index theorem in differential \(K\)-theory. Geom. Topol. 14(2), 903–966 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Harris, B.: Differential characters and the Abel-Jacobi map, Algebraic \(K\)-theory: connections with geometry and topology (Lake Louise, AB, 1987), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 279, pp. 69–86. Kluwer Academic Publishers, Dordrecht (1989)

  17. Harvey, R., Lawson, B., Zweck, J.: The de Rham-Federer theory of differential characters and character duality. Am. J. Math. 125(4), 791–847 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Harvey, R., Zweck, J.: Divisors and Euler sparks of atomic sections. Indiana Univ. Math. J. 50(1), 243–298 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hopkins, M., Singer, I.: Quadratic functions in geometry, topology, and M-theory. J. Differ. Geom. 70, 329–425 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Mathai, V., Quillen, D.: Superconnections, Thom classes, and equivariant differential forms. Topology 25(1), 85–110 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mei, X.M.: The transgression of Pontrjagin characteristic classes. Acta Math. Sin. 23(1), 37–45 (1980)

    MathSciNet  MATH  Google Scholar 

  22. Mei, X.M.: The integral formulas of Chern characteristic forms. J. Math. Res. Expo. (English Ed.) (1), 10–13 (1986)

  23. Takizawa, S.: On the Stiefel characteristic classes of a Riemannian manifold. Mem. Coll. Sci. Univ. Kyoto Ser. A Math. 28, 1–10 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wu, G.L.: The transgression of characteristic forms. I. Acta Math. Sin. 19(1), 52–62 (1976)

    MathSciNet  MATH  Google Scholar 

  25. Wu, G.L.: The transgression of characteristic forms. II. Acta Math. Sin. 19(2), 119–128 (1976)

    MathSciNet  MATH  Google Scholar 

  26. Zhang, W.: Lectures on Chern-Weil Theory and Witten Deformations, vol. 4. Nankai Tracts in Mathematics, World Scientific Publishing Co., Inc., River Edge (2001)

    MATH  Google Scholar 

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  1. Department of Mathematics, Hong Kong Baptist University, Kowloon, Hong Kong

    Man-Ho Ho

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  1. Man-Ho Ho
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Ho, MH. Gauss-Bonnet-Chern Theorem and Differential Characters. Results Math 72, 573–584 (2017). https://doi.org/10.1007/s00025-017-0652-1

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  • DOI: https://doi.org/10.1007/s00025-017-0652-1

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