关于谱不变量的一个注记(英文)

A NOTE ON SPECTRAL INVARIANTS

对于黎曼流形Ｍ，Ｍ′，证明了：如果对３个独立的ｐ值有ｓｐｅｃｐ（Ｍ）＝ｓｐｅｃｐ（Ｍ′），那么∫Ｍｒ２＊１＝∫Ｍ′ｒ′２＊１，∫Ｍ‖Ｒｉｃ‖２＊１＝∫Ｍ′‖Ｒｉｃ′‖２＊１，∫Ｍ‖Ｒｉｅｍ‖２＊１＝∫Ｍ′‖Ｒｉｅｍ′‖２＊１．

Let M and M′ be two Riemann manifolds. If  spec p(M)= spec p(M′) for some p, then which geometrical properties of M and M′ are the same? This paper discusses this problem and proves that if  spec p(M)= spec p(M′) for three independent values of p, then ∫ Mr 21=∫ M′ r′ 21, ∫ M‖ Ric ‖ 21=∫ M′ ‖ Ric′ ‖ 21 and ∫ M‖ Riem ‖ 21=∫ M′ ‖ Riem′ ‖ 21.