Lefschetz fixed point formula on a compact Riemannian manifold with   boundary for some boundary conditions
  In [8] the authors introduced a pair of new de Rham complexes on a compact oriented Riemannian manifold with boundary by using a pair of new boundary conditions to discuss the refined analytic torsion on a compact manifold with boundary. In this paper we discuss the Lefschetz fixed point formula on these complexes with respect to a smooth map having simple fixed points and satisfying some special condition near the boundary. For this purpose we are going to use the heat kernel method for the Lefschetz fixed point formula. 
Rung-Tzung Huang, Yoonweon Lee
math.DG