  The topological property of a gapped odd-parity superconductor is jointly
determined by its pairing nodes and Fermi surfaces in normal state. We reveal
that the contractibility of Fermi surfaces without crossing any time-reversal
invariant momentum and the presence of nontrivial Berry phase on Fermi surfaces
are two key conditions for the realization of higher-order topological
odd-parity superconductors (HOTOPSCs). When the normal state is a normal metal,
we reveal the necessity of removable Dirac pairing nodes and provide a general
and simple principle to realize HOTOPSCs. Our findings can not only be applied
to analyze the topological property of odd-parity superconductors, but also be
used as a guiding principle to design new platforms of higher-order topological
superconductors, as well as higher-order topological insulators owing to their
direct analogy in Hamiltonian description.
