Ricci张量对称函数的预定问题

The Prescribing Problem for Symmetric Function of Ricci Tensor

本文考虑Ricci张量的对称函数σ2 (Ricg)的预定问题.假设(M,g)是闭的Einstein流形,我们得到了只要流形(M,g)不具有σ2 (Ric)奇性,则对于变号的函数f∈C^∞(M),存在度量g^*,使得σ2 (Ricg^*)=f.然后,作为推论,得到了具有负数量曲率的闭Einstein流形上的预定曲率的结果.

We consider the prescribing problem for symmetric function of Ricci tensor.Suppose a closed Einstein manifold(M,g) is not σ2(Ric) singular.Let f∈C^∞(M) and it changes sign.We prove that there exists a metric g^* such that σ2(Ricg^*)=f.Then,as a corollary,we have an existence result for the prescribing problem for Einstein manifold with negative scalar curvature.