  We study bounded pseudoconvex domains in complex Euclidean spaces. We find
analytical necessary conditions and geometric sufficient conditions for a
domain being of trivial Diederich--Forn\ae ss index (i.e. the index equals to
1). We also connect a differential equation to the index. This reveals how a
topological condition affects the solution of the associated differential
equation and consequently obstructs the index being trivial. The proofs relies
on a new method of study of the complex geometry of the boundary. The method
was motivated by geometric analysis of Riemannian manifolds. We also generalize
our main theorems under the context of de Rham cohomology.
