  D.Freed has formulated and proved an index theorem on odd dimensional spin
manifolds with boundary. The proof is based on analysis by Calderon and Seeley.
In this note we are going to give a proof of this theorem using the heat
kernels methods for boundary conditions of Dirichlet and Von Neumann type.
Moreover we consider also the Atiyah-Patodi-Singer spectral boundary condition
which is not considered in Freed's paper. As a direct consequence of the
method, we will obtain some information about isospectral invariants of the
boundary conditions. This proof does not uses the cobordism invariance of index
and are easily generalized to family case.
