You could say "deformed" relative to "undeformed" or relative to
"original". The Euclidean space may be considered undeformed.

Not in GR, and not in geometry in general. In particular, it is not in
general possible to map the points of a Euclidean manifold 1-to-1 to the
points of a curved manifold, while preserving the local topological
structure (i.e. the neighborhoods).

This is so basic it is mind-boggling that you think this: try to
describe the "deformation" you imagine from E^2 to S^2 (infinite 3-d
Euclidean space to the surface of a sphere).... Then try a Klein
bottle.... These examples don't begin to touch the difference between
E^4 and E^(1,3), and are vastly simpler than the typical manifolds of GR....

Trying to consider "deformations" like this is hopeless. The geometry of
any manifold used as a physical model of the world must be understood
INTRINSICALLY.

IOW: in GR it is the distributions of matter and energy (plus boundary
conditions) that determine the geometry (topology and metric) of the
manifold; it is not possible to remove the matter and energy from the
universe -- but that is what must happen to get your "undeformed space"
(which isn't Euclidean, anyway). Removing all matter and energy from the
universe -=-

-=- yields a _DIFFERENT_ physical situation and a _DIFFERENT_
universe -- the word "deformed" is woefully inadequate to describe that.

IOW: you can only deform one thing to another if they are commensurate,
and different manifolds in general are not.

You may
also view the term "deformed" in the sense of "altered".

Hmmm. That's still woefully inadequate. <shrug>

Consider the
relative positions of N number of neighborhood points in the
Euclidean space. If the relative positions of all these points remain
"unchanged" or "unaltered" or the "same" for all time to come, that
is to say if the relative positions of all neighborhood points remain
"invariant" with time, we can say the space continuum is undeformed.
If on the other hand the relative positions of all these neighborhood
points get "changed" or "altered" at some later time due to 'some
reason', we can say the space continuum is deformed.

But that is not sufficient to handle the general case of a curved
manifold. In particular, this cannot alter the topology of the manifold,
as you are only admitting continuous deformations.

For instance, you cannot possibly obtain the Schwarzschild spacetime
from E^4 this way. In fact, you cannot even get Minkowski spacetime from
E^4 this way, and it _has_ the same topology as E^4.

Moreover, you are discussing evolution over time, which cannot possibly
obtain the manifolds of GR from Euclidean manifolds.

Specifically: you are imposing a geometrical structure E^3xR on the
world, where E^3 is "space" and R is "time". Such background structure
is clearly unwarranted -- why assume this particular structure rather
than some other?? GR, of course, avoids this by not having any
background geometrical structure at all (geometry is a dynamical aspect
of the model).

IOW: you are essentially assuming a Galilean world.

In our sincere and most revolutionary attempt to represent the
gravitational phenomenon through geodesic equations in four
dimensional space-time manifold, we mainly focused on the geometry of
particle traces in the space-time continuum.

That is not "revolutionary", that is quite standard in GR, which is ~90
years old. And your claim simply is not true -- most people "focus" on
the curvature and metric tensors, and the topological structure of the
manifold, not "particle traces".

In the process we lost
focus on the geometry of space continuum itself which by definition
consists of a continuum of points.

_YOU_ have indeed lost that (if you ever had it), but people
knowledgeable about GR have not. <shrug>

The geometry is wholly determined by the metric and the topology --
that's why people concentrate on them. <shrug>

Once we focus on the *geometry* of
the space continuum, the very first unique feature we notice is that
the space continuum itself gets deformed in the presence of
gravitational field.

I repeat: your usage of "deformed" is nonsense. The time evolutions you
call "deformations" are not at all what happens in GR. In GR the many
solutions to the field equation are each their own individual manifold,
and there is no direct relationship between any pair of them (in
general). None of them, of course, are Euclidean. <shrug>

Like all too many people around here, you need to STUDY GR and
differential geometry. Yes, to do that you will need to unlearn many
misconceptions.

[You seem to be resonating with Ken Tucker. I caution against
that, as he is in the same boat (needing to unlearn many
misconceptions). The two of you will sail off into never-
never land, unrelated to any physics, the real world, or
the _real_ theory of GR.]

Tom Roberts