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The Secondary Chern–Euler Class for a General Submanifold

  • Zhaohu Nie (a1)
Abstract

We define and study the secondary Chern–Euler class for a general submanifold of a Riemannian manifold. Using this class, we define and study the index for a vector field with non-isolated singularities on a submanifold. As an application, we give conceptual proofs of a result of Chern.

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References
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[1] Allendoerfer, C. B., The Euler number of a Riemann manifold. Amer. J. Math. 62(1940), 243248. http://dx.doi.org/10.2307/2371450
[2] Chern, S., A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds. Ann. of Math. 45(1944), 747752. http://dx.doi.org/10.2307/1969302
[3] Chern, S., On the curvatura integra in a Riemannian manifold. Ann. of Math. 46(1945), 674684. http://dx.doi.org/10.2307/1969203
[4] Chern, S. S. and Simons, J., Characteristic forms and geometric invariants. Ann. of Math. 99(1974), 4869. http://dx.doi.org/10.2307/1971013
[5] Fenchel, W., On total curvatures of Riemannian manifolds. I. J. London Math. Soc. 15(1940), 1522. http://dx.doi.org/10.1112/jlms/s1-15.1.15
[6] Morse, M., Singular points of vector fields under general boundary conditions. Amer. J. Math. 51(1929), no. 2, 165178. http://dx.doi.org/10.2307/2370703
[7] Nie, Z., Secondary Chern-Euler forms and the Law of Vector Fields. arXiv:0909.4754.
[8] Sha, J.-P., A secondary Chern-Euler class. Ann. of Math. 150(1999), no. 3, 11511158. http://dx.doi.org/10.2307/121065
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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