de　Rham－Hodge－Signature算子的局部指标定理

Local Index Theorem for de RHam-Hodge-Signature Operator

整体的Ａｔｉｙａｈ－Ｓｉｎｇｅｒ指标定理对于一般的椭圆微分算子都成立．但对于局部指标定理来说，人们只能对具体的算子给予证明．并且各种情况需个别处理．由［６］知本文中证得的ｄｅＲｈａｍ－Ｈｏｄｇｅ－Ｓｉｇｎａｔｕｒｅ算子的局部指标既不是α型也不是β型的．这和经典的椭圆算子情形不同．

There holds the global Atiyah-Singer Index Theorem for general elliptical differential operators. But for local index theorem,people can only give a proof for concrete operator and the teclnique used in the proof varies from one to another.We present here the suitable definition of de Rham-Hodge-Signature operator D and the proof of its locaf index theorem.From[6],we conclude that the local index for D is neither of type α nor type β. This is dtherent from the case of classical elliptic differential operators.