谱对局部对称黎曼流形的影响

THE INFLUENCE OF THE SPECTRUM ON A LOCALLY SYMMETRIC RIEMANNIAN MANIFOLD

用Ｓｐｅｃ￣ｑ表示黎曼流形上拉普拉斯算子作用在ｑ次形式上的谱。假设两个黎曼流形（Ｍ，ｇ）和（Ｍ’ｇ’）有相同的谱Ｓｐｅｃ￣０和Ｓｐｅｃ￣１．本文证明：ｉ）若（Ｍ，ｇ）是局部对称的共形平坦流形，则（Ｍ’，ｇ’）也是；ｉｉ）当（Ｍ，ｇ）和（Ｍ’，ｇ’）都是Ｋａｅｈｌｅｒ流形时，若（Ｍ，ｇ）是局部对称的Ｂｏｃｈｎｅｒ－Ｋａｅｂｌｅｒ流形，则（Ｍ’ｇ’）也是；进而，两种情况下，（Ｍ，ｇ）和（Ｍ’，ｇ’）都局部等距。

By Spec￣q we denote the spectrum of the Laplace operator acting on the space of q-forms on a Riemannian manifold.Assume that two Riemannian manifolds M and M' have thesame spectra Spec￣0 and Spec`1.It is proven thatⅠ)if M is locally symmetric and conformally flat,then so is M'.Ⅱ)when M and M' are Kaehler manifolds, if M is a locally symmetric Bochner-Kaehler mani-fold, then so is M’;moreover,in either case M and M' are locally isometric each other.