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Positive curvature, symmetry, and topology
Manuel AMANN
Front. Math. China. 2016, 11 (5): 1099-1122.
https://doi.org/10.1007/s11464-016-0580-5
We depict recent developments in the field of positive sectional curvature, mainly, but not exclusively, under the assumption of isometric torus actions. After an elaborate introduction to the field, we shall discuss various classification results, before we provide results on the computation of Euler characteristics. This will be the starting point for an examination of more involved invariants and further techniques. In particular, we shall discuss the Hopf conjectures, related decomposition results like the Wilhelm conjecture, results in differential topology and index theory as well as in rational homotopy theory, geometrically formal metrics in positive curvature and much more. The results we present will be discussed for arbitrary dimensions, but also specified to small dimensions. This survey article features mainly depictions of our own work interest in this area and cites results obtained in different collaborations; full statements and proofs can be found in the respective original research articles.
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Gauss-Bonnet-Chern mass and Alexandrov-Fenchel inequality
Yuxin GE,Guofang WANG,Jie WU,Chao XIA
Front. Math. China. 2016, 11 (5): 1207-1237.
https://doi.org/10.1007/s11464-016-0558-3
This is a survey about our recent works on the Gauss-Bonnet-Chern (GBC) mass for asymptotically flat and asymptotically hyperbolic manifolds. We first introduce the GBC mass, a higher order mass, for asymptotically flat and for asymptotically hyperbolic manifolds, respectively, by using a higher order scalar curvature. Then we prove its positivity and the Penrose inequality for graphical manifolds. One of the crucial steps in the proof of the Penrose inequality is the use of an Alexandrov-Fenchel inequality, which is a classical inequality in the Euclidean space. In the hyperbolic space, we have established this new Alexandrov-Fenchel inequality. We also have a similar work for asymptotically locally hyperbolic manifolds. At the end, we discuss the relation between the GBC mass and Chern’s magic form.
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On combinatorial Gauss-Bonnet Theorem for general Euclidean simplicial complexes
Stephan KLAUS
Front. Math. China. 2016, 11 (5): 1345-1362.
https://doi.org/10.1007/s11464-016-0575-2
For a finitely triangulated closed surface M2, let αx be the sum of angles at a vertex x. By the well-known combinatorial version of the 2-dimensional Gauss-Bonnet Theorem, it holds ∑x(2π−αx) = 2πχ(M2), where χ denotes the Euler characteristic of M2, αx denotes the sum of angles at the vertex x, and the sum is over all vertices of the triangulation. We give here an elementary proof of a straightforward higher-dimensional generalization to Euclidean simplicial complexes K without assuming any combinatorial manifold condition. First, we recall some facts on simplicial complexes, the Euler characteristics and its local version at a vertex. Then we define δ(τ) as the normed dihedral angle defect around a simplexτ. Our main result is ∑τ (−1)dim(τ)δ(τ) =χ(K), where the sum is over all simplices τ of the triangulation. Then we give a definition of curvature κ(x) at a vertex and we prove the vertex-version ∑x∈K0κ(x) =χ(K) of this result. It also possible to prove Morse-type inequalities. Moreover, we can apply this result to combinatorial (n + 1)-manifolds W with boundary B, where we prove that the difference of Euler characteristics is given by the sum of curvatures over the interior of W plus a contribution from the normal curvature along the boundary B:χ(W) −12χ(B) = ∑τ∈W−B(−1)dim(τ)δ(τ) +∑τ∈B(−1)dim(τ)ρ(τ).
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13 articles
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