Manifolds of Riemannian metrics with prescribed scalar curvature
Manifolds of Riemannian metrics with prescribed scalar curvature
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具有规定标量曲率的黎曼度量流形
DOI：
10.1090/s0002-9904-1974-13457-9
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发表时间：
1974
影响因子：
1.3
通讯作者：
J. Marsden
中科院分区：
文献类型：
--
作者：
A. Fischer;J. Marsden
定理2.假设J * V为0。写作UtQ=（jeao\0）U&9J（\）是闭子流形的不交并。注：如果dim M = 2，ej = 1 - 8，并且如果dim M = 3，则可以丢弃1F* Je 0的假设。定理1的证明还允许我们得出结论，线性化方程DR（g 0）· h=0的解h通过给定的解g 0与R（g）=p的精确解的曲线相切，只要p不是常数^ 0。在[4]的术语中，我们说方程R（g）=p在g 0是线性化稳定的。根据下面的定理3，如果Ric（g 0）不恒为零，则方程R（g）=0仍然关于解g 0线性化稳定。对于奇异情形p = 0，定理2包含了一个受Brill和Deser [2]工作启发的孤立定理，即平坦度量是R（g）=0的孤立解.作为一个推论，我们有：如果g（t）是一个
THEOREM 2. Assume J * V 0 . Writing UtQ=(je a o\0 )U&9 J(\ is the disjoint union of closed submanifolds. REMARK. If d i m M = 2 , e^J=^" 8 , and if d i m M = 3 , the hypothesis that 1F*J£0 can be dropped. The proof of Theorem 1 also allows us to conclude that a solution h of the linearized equations DR(g0) • h=0 is tangent to a curve of exact solutions of R(g)=p through a given solution g0, provided p is not a constant ^ 0 . In the terminology of [4] we say the equation R(g)=p is linearization-stable at g0. From Theorem 3 below the equation R(g)=0 is still linearization-stable about a solution g0 provided Ric(g0) is not identically zero. For the singular case p = 0 , Theorem 2 incorporates an isolation theorem inspired by the work of Brill and Deser [2], namely, that the flat metrics are isolated solutions of R(g)=0. As a corollary one has: If g(t) is a