论文标题
积分几何霍夫猜想
Integral geometric Hopf conjectures
论文作者
论文摘要
HOPF标志指出，正曲率的紧凑riemannian 2d-manifold m具有Euler特征X（M）> 0，并且在负曲率x（m）（-1）（-1）^d> 0的情况下。 HOPF产品猜想询问是否可以在S^2 X S^2等产品歧管上存在一个正曲率度量。通过以几何形式制定曲率积分，可以探索这些问题的有限简单图，从而导致线性编程问题。在此更大的说明文件中，我们旨在探索Hopf猜想的历史，并提及一些已尝试的攻击策略。我们通过证明在每个正弯曲歧管中都存在一个新的积分理论MU形成概念，其中存在一个满足高斯bonnet-chern x（m）= \ int_m k dv的mu狂热k，因此k在一个开放的体积u上是积极的。
The Hopf sign conjecture states that a compact Riemannian 2d-manifold M of positive curvature has Euler characteristic X(M)>0 and that in the case of negative curvature X(M) (-1)^d >0. The Hopf product conjecture asks whether a positive curvature metric can exist on product manifolds like S^2 x S^2. By formulating curvature integral geometrically, these questions can be explored for finite simple graphs, where it leads to linear programming problems. In this more expository document we aim to explore also a bit of the history of the Hopf conjecture and mention some strategies of attacks which have been tried. We illustrate the new integral theoretic mu-curvature concept by proving that for every positive curvature manifold M there exists a mu-curvature K satisfying Gauss-Bonnet-Chern X(M)=\int_M K dV such that K is positive on an open set U of volume arbitrary close to the volume of M.
