What is recorded here?
Chern gives an intrinsic differential-form proof that integrates curvature to the Euler characteristic for closed oriented even-dimensional Riemannian manifolds, avoiding the earlier proof's use of local Euclidean embeddings. Local curvature and the overall shape of a space can appear to be unrelated. This proof connects them by lifting the problem to the space of unit tangent directions. There, a curvature form becomes the derivative of another form. A vector field with an isolated singularity lets Stokes' theorem turn the integral into a count expressed by that singularity's index. The equality was already known in general closed cases; Chern's contribution here is an intrinsic route to it. The paper offers a clear method for connecting geometric quantities and topological invariants, rather than claiming every later characteristic-class result.
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Scientific scope
Exact 1944 Annals 45(4),747-752 original: all six article pages/PDF 2-7 text and footnotes read, PDF 1 archive cover metadata only. Closed orientable even-dimensional Riemannian manifold, with the differentiability/metric regularity stated in the paper; no boundary/noncompact/odd-dimensional or lowered-regularity extension imported. Curvature top form is pulled back to the unit tangent sphere bundle, where transgression makes it exact; it is not claimed exact on the closed base. Vector-field index theorem and Stokes are inherited. General formula already attributed in the introduction to Allendoerfer/Weil; Chern supplies an intrinsic proof, not sole first discovery of all cases. Later polyhedral extensions, all Chern/characteristic classes, index theory and downstream applications separate. The complete symbolic proof and its coefficients have not been independently reconstructed or certified here. independent scientific/calibration certification NONE.
Read the original work
- Chern, A Simple Intrinsic Proof of the Gauss-Bonnet Formula for Closed Riemannian Manifolds, Annals45(4),747-752(1944), original scan
- Publisher-deposited Crossref identity for Chern1944 intrinsic proof
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People and systems
Shiing-Shen Chern
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English catalogue narrative
Chern: an intrinsic proof connecting curvature to Euler characteristic
Chern gives an intrinsic differential-form proof that integrates curvature to the Euler characteristic for closed oriented even-dimensional Riemannian manifolds, avoiding the earlier proof's use of local Euclidean embeddings.
Local curvature and the overall shape of a space can appear to be unrelated. This proof connects them by lifting the problem to the space of unit tangent directions. There, a curvature form becomes the derivative of another form. A vector field with an isolated singularity lets Stokes' theorem turn the integral into a count expressed by that singularity's index. The equality was already known in general closed cases; Chern's contribution here is an intrinsic route to it. The paper offers a clear method for connecting geometric quantities and topological invariants, rather than claiming every later characteristic-class result.
Scientific scope
Exact 1944 Annals 45(4),747-752 original: all six article pages/PDF 2-7 text and footnotes read, PDF 1 archive cover metadata only. Closed orientable even-dimensional Riemannian manifold, with the differentiability/metric regularity stated in the paper; no boundary/noncompact/odd-dimensional or lowered-regularity extension imported. Curvature top form is pulled back to the unit tangent sphere bundle, where transgression makes it exact; it is not claimed exact on the closed base. Vector-field index theorem and Stokes are inherited. General formula already attributed in the introduction to Allendoerfer/Weil; Chern supplies an intrinsic proof, not sole first discovery of all cases. Later polyhedral extensions, all Chern/characteristic classes, index theory and downstream applications separate. The complete symbolic proof and its coefficients have not been independently reconstructed or certified here. independent scientific/calibration certification NONE.
People and systems
Shiing-Shen Chern