Abstract. The Dirac operators are important differential operators. Bismut got a local index theorem for the modified Dirac operators over the Spin manifolds. In this paper, we obtained a similar theorem over the Spin$^c$-manifolds and used the heat equation to prove this theorem.
AMS Subject Classification
Received March 18, 2004, and in revised form April 16, 2005. (Registered under 5876/2009.)