Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Gauss-Bonnet-Chern Theorem on Riemannian Manifolds   This expository paper contains a detailed introduction to some important
works concerning the Gauss-Bonnet-Chern theorem. The study of this theorem has
a long history dating back to Gauss's Theorema Egregium (Latin: Remarkable
Theorem) and culminated in Chern's groundbreaking work [14] in 1944, which is a
deep and wonderful application of Elie Cartan's formalism. The idea and tools
in [14] have a great generalization and continue to produce important results
till today. In this paper, we give four different proofs of the
Gauss-Bonnet-Chern theorem on Riemannian manifolds, namely Chern's simple
intrinsic proof, a topological proof, Mathai-Quillen's Thom form proof and
McKean-Singer-Patodi's heat equation proof. These proofs are related with
remarkable developments in differential geometry such as the Chern-Weil theory,
theory of characteristic classes, Mathai-Quillen's formalism and the
Atiyah-Singer index theorem. It is through these brilliant achievements the
great importance and influence of Chern's insights and ideas are shown. Our
purpose here is to use the Gauss- Bonnet-Chern theorem as a guide to expose the
reader to some advanced topics in modern differential geometry.

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The categories of this paper are: cmp-lg
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The categories of this paper are: dg-ga
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The categories of this paper are: funct-an
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The categories of this paper are: bayes-an
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The categories of this paper are: cs
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The categories of this paper are: eess
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The categories of this paper are: q-bio
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The categories of this paper are: acc-phys
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The categories of this paper are: comp-gas
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The categories of this paper are: mtrl-th
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The categories of this paper are: adap-org
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The categories of this paper are: atom-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: q-alg
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The categories of this paper are: supr-con
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The categories of this paper are: solv-int