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Curvature in Riemannian Manifolds

2020
Since the notion of curvature can be defined for curves and surfaces, it is natural to wonder whether it can be generalized to manifolds of dimension n ≥ 3. Such a generalization does exist and was first proposed by Riemann. However, Riemann’s seminal paper published in 1868 two years after his death only introduced the sectional curvature, and did not
Jean Gallier, Jocelyn Quaintance
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On Gaussian and Geodesic Curvature of Riemannian Manifolds

Canadian Journal of Mathematics, 1974
In [1], S. S. Chern gave a very elegant and simple proof of the Gauss-Bonnet formula for closed (i.e. compact without boundary) oriented Riemannian manifolds of even dimension:Here, c is a suitable constant depending on the dimension of M and Ω is an n-form (n = dim M) which may be calculated from its curvature tensor. W.
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