Use of Invariant metric/distances to classify domains in $\mathbb{C}^n$ I am a graduate student in mathematics, who works usually in operator theory. Lately I had to read about about the Lempert’s theorem(a theorem regarding when some pseudometric/distances coincide)  and the domains for which the theorem holds. I wanted to know why actually does one study the same.
It is said on the Introduction  of this thesis ‘In order to study domains in $n$-dimensional complex space $\mathbb{C}^n$, it is important to associate with these domains distance functions which are invariant under biholomorphic mappings. Such functions can then be used as a tool to determine whether domains in $\mathbb{C}^n$ are biholomorphically equivalent.’
Again the Introduction in this thesis says that
‘Biholomorphically invariant metrics appear to be an important tool in Several Complex Variables which generalize the concept of Hermitian metrics. The Carath ́eodory and Kobayashi pseudometrics are instances of such metrics; both generalize the Poincar ́e metric and play an important role in the classification of domains’.
Can anyone tell ‘how’ actually are such invariant distances  used as a tool to determine whether domains in $\mathbb{C}^n$ are biholomorphically equivalent?
Because everywhere they just mention the above and don’t actually explain how. Can you explain it in simple functional analysis terms? As I am not much familiar with differential/ hyperbolic  geometry.
 A: The best way to answer you is perhaps to directly quote part of the introduction of the beautiful book "Invariant distances and metrics in complex analysis" by M. Jarnicki and P. Pflug.
Here is the quote:
"One of the most beautiful results in the classical complex analysis is the Riemann
mapping theorem which says that, except the whole complex plane, every simply
connected plane domain is biholomorphically equivalent to the unit disc. Thus,
the topological property "simply connected" is already sufficient to describe, up to
biholomorphisms, a large class of plane domains. On the other hand, the Euclidean
ball and the bidisc in $\mathbb C^2$ are topologically equivalent simply connected domains but
they are not biholomorphic. This observation, which was made by H. Poincaré as
early as at the end of the last century, shows that even inside the class of bounded
simply connected domains there is no single model (up to biholomorphisms) as it
is in the plane case. Therefore, it seems to be important to associate with domains
in $\mathbb C^n$ tractable objects that are invariant under biholomorphic mappings. Provided
that these objects are sufficiently concrete, one can hope to be able to decide, at
least in principle, whether two given domains are biholomorphically distinct.
An object of this kind was introduced, for example, by C. Caratheodory in the
thirties. His main idea was to use the set of bounded holomorphic functions as an
invariant. More precisely, he defined pseudodistances on domains via a "generalized" Schwarz Lemma. A specific property of these pseudodistances is that holomorphic
mappings act as contractions. Thus, in particular, biholomorphic mappings operate
as isometries. For such objects the name "invariant pseudodistances" has become
very popular. This is where the title of our book comes from, although in the
text we prefer to talk about holomorphically contractible pseudodistances. Apart
from the class of bounded holomorphic functions, other classes of functions are
used to obtain, via extremal problems, new objects contractible with respect to
certain families of holomorphic mappings. For example, the class of square
integrate holomorphic functions was used by S. Bergman. Moreover, all these objects
admit infinitesimal versions associating to any "tangent vector" a specific length
contractible under holomorphic mappings. Besides using families of functions to
associate (via an extremal problem) tractable objects with domains in $\mathbb C^n$, one can
consider sets of analytic discs as new biholomorphic invariants. This idea is due
to S. Kobayashi."
Summing up, I would rather say that these invariant distances can be used as a tool to determine whether domains in $\mathbb C^n$ are biholomorphically inequivalent!
For instance, to biholomorphically distinguish the unit ball $\mathbb B^n$ in $\mathbb C^n$ and the polydisc $\Delta^n$ one can consider the Bergman metrics of these two domains, compute their holomorphic sectional curvatures and observe that it is constant for $\mathbb B^n$, while it is not constant for $\Delta^n$.
