Remarks on curvature and the Euler integrand
Remarks on curvature and the Euler integrand
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关于曲率和欧拉被积函数的评论
DOI：
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发表时间：
1971
期刊：
影响因子：
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通讯作者：
A. Weinstein
中科院分区：
文献类型：
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作者：
A. Weinstein
将被称为2n维的欧拉被积函数，因为通过广义高斯-邦纳定理，通过在正交标架中的曲率张量的分量上计算χ并使用与给定黎曼度量相关联的体积元对所得的M上的真实的值函数积分，获得了2n维定向黎曼流形M的欧拉特征线，直到正常数。它已被H. Hopf证明了具有正截面曲率的偶数维黎曼流形的Euler特征是正的，甚至在这种情况下Euler被积函数也可能是正的.本说明专门就这一问题提出一些意见。我们继续修正一些术语。多项式函数σ：K x R x R -> R由下式定义：
will be called the Euler integrand in dimension 2n since, by the generalized Gauss-Bonnet theorem, the Euler characteristic of an oriented riemannian manifold M of dimension 2n is obtained, up to a positive constant, by evaluating χ on the components of the curvature tensor in orthonormal frames and integrating the resulting real valued function over M, using the volume element associated with the given riemannian metric. It has been conjectured by H. Hopf that the Euler characteristic of an even dimensional riemannian manifold with positive sectional curvature is positive, and it may even be the case that the Euler integrand is positive in this situation. The present note is devoted to the presentation of some remarks on this question. We continue by fixing some more terminology. The polynomial function σ: K x R x R -> R defined by the formula