  In this paper, we consider an interior transmission eigenvalue problem on two
compact Riemannian manifolds with common smooth boundary.
  We suppose that a couple of these manifolds is equipped with locally
anisotropic type Riemannian metric tensors, i.e., these two tensors are not
equivalent in a neighborhood of common boundary.
  Here we note that we do not assume that these manifolds are diffeomorphic.
  In addition, we impose some conditions of the refractive indices in a
neighborhood of common boundary.
  Then we prove that the set of ITEs form infinite discrete set and the
existence of ITE-free region.
  In order to prove our results, we employ so-called the $T$-coercivity method.
