  Study of symmetry, topology and geometric phase can reveal many new and
interesting results on the topological states of matter. Here we present a
completely new and interesting result of symmetry, topology and quantization of
geometric phase along with the physical explanation for two different symmetry
classes. We present a detailed study of the auxiliary space for two different
symmetry classes of Hamiltonians. We show explicitly that the origin of the
auxiliary space inside the curve is only a necessary condition but it is not a
sufficient condition for the topological state. One of the most interesting
results is that same symmetry-class Hamiltonians show different behaviour in
topology and quantized geometric phase.
