  Topological phases are generally characterized by topological invariants
denoted by integer numbers. However, different topological systems often
require different topological invariants to measure, such as geometric phases,
topological orders, winding numbers, etc. Moreover, geometric phases and its
associated definitions usually fail at critical points. Therefore, it's
challenging to predict what would occur during the transformation between two
different topological phases. To address these issues, in this work, we propose
a general definition based on fidelity and trace distance from quantum
information theory: manifold distance. This definition does not rely on the
berry connection of the manifolds but rather on the information of the two
manifolds - their ground state wave functions. Thus, it can measure different
topological systems (including traditional band topology models, non-Hermitian
systems, and topological order models, etc.) and exhibit some universal laws
during the transformation between two topological phases. Our research
demonstrates that when the properties of two manifolds are identical, the
distance and associated higher-order derivatives between them can smoothly
transition to each other. However, for two different topological manifolds, the
higher-order derivatives exhibit various divergent behaviors near the critical
points. For subsequent studies, we expect the method to be generalized to
real-space or non-lattice models, in order to facilitate the study of a wider
range of physical platforms such as open systems and many-body localization.
