Since no answer is provided thus far, I will attempt at a tentative answer along geometric lines.
The answer is based on EPH-classifications in Geometry, Algebra, Analysis and Arithmetic and exposes a basic trichotomy in Geometry, Algebra, Analysis and Arithmetic between Elliptic equations/spaces, Parabolic equations/spaces and Hyperbolic equations/spaces (that goes back to ancient greek mathematics on classifications of conic sections).
Specificaly:
Trichotomy of Elliptic-Parabolic-Hyperbolic appears in many different
areas of mathematics. All of these are named after the very first
example of trichotomy, which is formed by ellipses, parabolas, and
hyperbolas as conic sections. We try to understand if these
classifications are justified and if similar mathematical phenomena is
shared among different cases EPH-classification is used.
At first glance EPH-classification is nothing deep. A discriminant of
a quadratic form being negative, zero or positive should not be that
important for people to make so much fuss about it. Geometric
similarity between the zeros of quadratic forms in affine space, could
be a motivation for naming them differently. But why is it that the
elliptic, parabolic, hyperbolic names are used so widely in
mathematics, in places where it seems irrelevant to the shape of
hyperbola, parabola and ellipse? The question is, does there exist a
paradigm behind these namings.
Of interest is that complex numbers are related to elliptic spaces, dual numbers (grassmann numbers) to parabolic spaces and usual numbers to hyperbolic spaces.
[..]We shall justify below why ”Elements of $SL_2(R)$” are named elliptic,
parabolic, and hyperbolic after the original paradigm of conic
sections. ”Actions of $SL_2(R)$ on complex numbers, double numbers, and
dual numbers” are also related to the geometry of conic sections.
[..]A key part of the study of Mobius transformations is their
classification into elliptic, parabolic and hyperbolic
transformations. The classification can be summarized in terms of the
fixed points of the transformation, as follows: Parabolic: The
transformation has precisely one fixed point in the Riemann sphere.
Hyperbolic: There are exactly two fixed points, one of which is
attractive, and one of which is repelling. Elliptic: There are exactly
two fixed points, both of which are neutral. Now we turn to studying
the connection between Mobius transformations and conic sections. The
transformations we consider are those that preserve the unit disk . It
is straightforward to see that all such Mobius transformations
correspond to matrices in $SU(1, 1)$. Given such a matrix , the Mobius
transform maps the unit disk into itself in such a way that it maps
arcs orthogonal to the unit circle to arcs orthogonal to the unit
circle. These transforms are in fact the isometries of the Poincare
disk.
[..]The $SL_2(R)$ action by Mobius transformations is usually considered as
a map of complex numbers $z = x + iy, i^2 = −1$. Moreover, this action
defines a map from the upper half-plane to itself. However there is no
need to be restricted to the traditional route of complex numbers
only. Less-known double and dual numbers also have the form $z = x + iy$
but different assumptions on the imaginary unit $i$: $i^2 = 0$ or $i^2 = 1$
correspondingly. Although the arithmetic of dual and double numbers is
different from the complex ones, e.g., they have divisors of zero,
we are still able to define their transforms by Mobius
transformations in most cases. Three possible values $-1, 0$, and $1$ of $\sigma = i^2$
will be referred to here as elliptic, parabolic, and hyperbolic
cases respectively.
[..]To understand the Mobius action in all EPH cases, we use the Iwasawa
de- composition of $SL_2(R) = ANK$ into three one-dimensional
subgroups $A, N, K$: $\begin{pmatrix}a & b\\\ c & d\end{pmatrix} = \begin{pmatrix}k & 0\\\ 0 & k-1\end{pmatrix} \begin{pmatrix}1 & w\\\ 0 & 1\end{pmatrix} \begin{pmatrix}Cos \theta & −Sin \theta\\\ Sin
> \theta & Cos \theta\end{pmatrix}$.
Subgroups $A$ and $N$ act in (1)
irrespective of the value of $\sigma$: $A$ makes a dilation by $α^2$,
i.e., $z \rightarrow α^2z$, and $N$ shifts points to left by $ν$, i.e.
$z \rightarrow z + ν$. By contrast, the action of the third matrix
from the subgroup $K$ sharply depends on $\sigma$. In the elliptic,
parabolic and hyperbolic cases K-orbits are circles, parabolas and
(equilateral) hyperbolas correspondingly.
[..][Riemann’s] uniformization theorem says that every simply
connected Riemann surface is conformally equivalent to one of the
three domains: the open unit disk, the complex plane, or the Riemann
sphere. In particular it admits a Riemannian metric of constant
curvature. This classifies Riemannian surfaces as elliptic (pos-
itively curved rather, admitting a constant positively curved metric),
parabolic (flat), and hyperbolic (negatively curved) according to
their universal cover. The uniformization theorem is a generalization
of the Riemann mapping theo- rem from proper simply connected open
subsets of the plane to arbitrary simply connected Riemann surfaces.
[..]From this, a classification of metrizable surfaces follows. A
connected metriz- able surface is a quotient of one of the following
by a free action of a discrete subgroup of an isometry group:
1.the sphere (curvature +1)
2.the Euclidean plane (curvature 0)
3.the hyperbolic plane (curvature −1).
