TITLE:
On Möbius Transformations as Polya Vector Fields from Lie Theory Viewpoint
AUTHORS:
Lucian M. Ionescu, Gabriel-Teodor Pripoae, Cristina-Liliana Pripoae
KEYWORDS:
Mobius Transformations, Polya Vector Fields, Lie Theory, 2D-Electromagnetism, Fluid Dynamics
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.16 No.9,
September
21,
2026
ABSTRACT: This expository article connects the conformal geometry of Möbius transformations with their kinematic interpretation as Pólya vector fields. We review the vector-calculus perspective of holomorphic functions, establishing a rigorous analytic-to-dynamic dictionary that maps complex analytic objects to conservative and solenoidal 2D physical flows. Leveraging the Lie theory of
S
L
2
(
ℂ
)
, we explicitly compute the one-parameter subgroups and their associated invariant vector fields, demonstrating how Möbius transformations can be understood dynamically as superpositions of elementary flows via the adjoint representation. Finally, as a motivational outlook for future research, we contextualize these geometric and Lie-theoretic results within a broader mathematical physics framework. We discuss heuristic analogies with 2D-electromagnetism, Hopf fibrations, and the Cayley-Dickson doubling construction as structural stepping stones toward higher-dimensional gauge theories.