From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Question about non-euclidian geometry
Date: 19 Aug 1996 22:01:50 GMT
In article <4usqsc$mnn@news.indy.net>, Greg Moriarty wrote:
>Usually to grasp the concept of non-euclidian geometry the student is told
>to imagine a 2 dimensional plane wrapped around into a 3 dimensional
>sphere. To the ant living on the plane, to which the 3rd dimension is a
>foreign concept, parallel lines seem to intersect and triangles add up to
>>180 degrees and other such silliness. To the god-like observer, those
>things aren't odd at all since the god-like creature has an understanding
>of the 3rd dimesion. My question is thus:
>
>Can one go from a non-euclidian geometry to a euclidian geometry by
>introducing a new dimension? Is this something that is done? If so, can
>one ALWAYS do so?
and later...
>On a side note, I remember a class called differential geometry
>in college. Does this deal with non-euclidean goemetry? Is
>N-E geometry a part of differential geometry? Are they synonymous?
You have to tell us just what "geometry" means. Differential geometry
is the study of Riemannian manifolds, roughly, sets which may locally
be deformed into a Euclidean space, and which allow the measurement of
lengths, but not perhaps in a way consistent with the deformation. Your
ant, above, thinks the portion of the sphere close to it looks like
a portion of the plane. The ant can also measure distances by walking
along curves, seeking the shortest possible path joining two points.
But a clever ant would note that the one-to-one correspondence between
its little neighborhood and a portion of the plane does not
preserve distances, angles, and lengths (hence the frustration of
global map-makers).
In this context, what you appear to be asking for is true: there is a
theorem of Nash (recent recipient of the Nobel Memorial prize in
Economics) which states that every (suitable) Riemannian manifold is
isometric with (= "the same as") a subset of some higher-dimensional
Euclidean space. (I think the dimension of the latter is quadratic
in the dimension of the original).
Note that Euclidean space R^n itself is also an Riemannian manifold
under the usual metric, so no, "differential geometry" does not
really presuppose "non-Euclidean geometry".
On the other hand, there are other interpretations of "geometry" for
which this discussion is inappropriate. For example, finite group
theorists often make use of "geometries", which are simply sets of
sets with certain incidence properties. In this direction I might
also mention the projective geometries (and affine geometries) which
more or less correspond to replacing the real numbers with other fields
(or rings) but still using coordinates; now "distance" is not
really defined.
dave
==============================================================================
From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Question about non-Euclidean geometry
Date: 20 Aug 1996 18:54:47 GMT
I would like to clarify a couple of responses to the original post,
which began:
In article <4usqsc$mnn@news.indy.net>, Greg Moriarty wrote:
>Usually to grasp the concept of non-euclidian geometry the student is told
>to imagine a 2 dimensional plane wrapped around into a 3 dimensional
>sphere. To the ant living on the plane, to which the 3rd dimension is a
>foreign concept, parallel lines seem to intersect and triangles add up to
>>180 degrees and other such silliness. To the god-like observer, those
>things aren't odd at all since the god-like creature has an understanding
>of the 3rd dimesion. My question is thus:
>
>Can one go from a non-euclidian geometry to a euclidian geometry by
>introducing a new dimension? Is this something that is done? If so, can
>one ALWAYS do so?
I responded to this (and some later points) in a post which might
appear to be at variance with another follow-up, which I feel sort of
misses the point.
In article <4vagvb$hpl@mountaindew.eng.umd.edu>,
Kevin Anthony Scaldeferri wrote:
>There is a theorem which says that any manifold satisfying the conditions
>(mumble,mumble) can be embedded in a Euclidean space of sufficiently
>large dimension. (Specifically 2N+1, where N is the dimension of the
>manifold in question). As I recall the conditions are not very
>stringent, possibly just that the manifold be differentiable. Any
>topology text should have the exact statement. (Whittney Embedding
>Theorem). I suggest Guillemin and Pollack as a fairly accessible book
>to find out more, if you are so inclined.
This gives an opportunity to explain a situation which arises quite
frequently in geometry and topology (and elsewhere in mathematics),
namely, what does it mean to call two things "the same"? Usually this
means that one can set up a one-to-one correspondence between the
points in the two objects, such that this correspondence (and its
inverse) "preserve the structure" of the things. But then one must ask
just what structure is to be preserved. (Keywords: Category Theory).
Let's begin with Moriarity's ant. The ant is convinced that it lives
on a 2-dimensional plane, and indeed demonstrates this by establishing
"rectangular" coordinates on the sphere -- usually, latitude and
longitude. All is well unless the ant is well-traveled: when
crossing the meridian opposite the Greenwich meridian, one coordinate
suddenly switches from +180 to -180 or vice versa. If the ant believed
that it lived on the plane, it would have to conclude it had suddenly
jumped from one edge of a rectangle to an opposite edge.
Clearly, then, our coordinates have not demonstrated that the ant
lives on the plane, but they are still useful. Indeed, even the ants
near this "intercolonial date-line" would have no trouble establishing
coordinates. They might simply agree to measure longitude from 0 to
360 -- from their perspective, it's the ants near Greenwich which have
trouble. Tourists from the two regions simply understand that when
traveling to the other part of the sphere, one changes coordinates by
adding or subtracting 360 degrees (in some cases) to the longitude.
There is a more serious problem near the poles, where latitude and
longitude reduce, more or less, to polar coordinates near the origin
in the plane. In particular, there is no longitude which can be
assigned to the poles in a way which would vary continuously near the
poles. Of course, residents near the poles can simply lay out a
rectangular grid of coordinates near the poles; they would have no
problem locating a point by giving it's coordinates "up from Main
Street and over from First Avenue". Again, tourists would have to be
willing to convert coordinates when traveling. This time the
change-of-coordinates calculations are more complex, requiring the
familiar trigonometric functions, but at least at all points where
both sets of coordinates are defined, the transformation rules are
differentiable functions.
With this set of four coordinate systems, the ants have collectively
proven that they live on what we call a _manifold_: a (Hausdorff)
topological space with the property that every point has a
neighborhood homeomorphic to Euclidean space. Moreover, their choice
of coordinate systems has the property that the change-of-coordinates
functions may be differentiated k times (with continuous k-th
derivatives) for any k = 0, 1, 2, ..., (even k=infinity, that is, the
functions are real-analytic). Thus we say that the manifold they have
set up is a _differentiable manifold_ or a _C^k-manifold_.
One easy way to get a manifold is to take a subset M of Euclidean space R^n
which has the property that near each point p of M one may choose a
new coordinate system for R^n such that the points of M which are near
p are precisely the points of R^n near p for which a few of the
coordinates are zero. We know, for example, that the unit sphere S inside
R^3 may be described in this way, if for all points on R^3 excluding
the origin we use spherical coordinates. (I suppose I have to use
r-1, theta, and phi, actually, to be consistent with the opening sentence:
then S is exactly the set of points where the first coordinate is zero.)
Manifolds created in this way are called _submanifolds of R^n_. Again one
can ask about the smoothness of the transition functions between the
new coordinates selected for neighborhoods of R^n, relative to the
standard coordinates; these define the smoothness of the submanifold.
Now to the Whitney theorem(s). Suppose the ants in various parts of a
planet managed to set up coordinate systems near their homes. Each
hangs a map up in the living room, showing "home" in the center, a big
square around it, and at the edges some darkness with the message
"Here be anteaters". A planetary conference is called, and amazingly,
each ant's map is found to overlap with it's neighbors' just a little,
and the coordinates each one uses can be converted to it's neighbors'
coordinates just by adding or subtracting a constant to one or both
coordinates (just as the Greenwich and anti-Greenwich crowds do). But
something unusual emerges: When the Greenwich and anti-Greenwich
crowds on earth superimpose their maps along the overlaps, they find a
portion of the earth repeated on the left and right edges -- once each
from each mapmaker's map. On the other hand, these ants notice that
when all the maps are superimposed, there is some repetition along all
four edges. But what shows up at the bottom left is repeated at the
top _right_; the rest of the left edge from bottom to top is repeated
on the right from top to bottom. Likewise, the top edge repeats the
bottom edge but with the orientation reversed.
The fact that Euclidean coordinates are established at each point
shows that the planet is a manifold (indeed, since the change-of-coordinate
maps are translations, it's a C^k manifold for all k). But it looks
different from the plane, and it doesn't appear to be the sphere either.
Indeed it's not: one can show that there is no continuous one-to-one
correspondence between the points on their planet and the points on the
sphere.
So what does the planet "look like"? A good picture might be found if
we could find a submanifold of Euclidean space to which this planet is
homeomorphic ("the same"). Some experimentation might lead you to
despair, since, as it turns out, there is _no_ submanifold of R^3
which is the same as this planet.
In 1935, Hassler Whitney showed that every C^infinity manifold M of
dimension n may be embedded in R^N as long as N > 2n, that is, there
is a submanifold of R^N whose points may be put into 1-to-1 correspondence
with M in a continuous way (with the inverse also continuous). Actually,
the proof can be made simpler if you don't care how big N is, but there's
some fascination with the minimal embedding dimension. Indeed, Whitney
expended considerable effort improving the theorem to say " ... N >= 2n "
in 1944.
In particular, our ants, living on an n=2 - dimensional planet
can view themselves as living on a subset of R^4. (Their manifold is known as
"Projective 2-space" or "the projective plane"). I'm sorry, I don't know
offhand a concise description of such a subset of R^4 in this way, but
here's an embedding of their planet into R^5: it's the subset defined by
the equations
(1-x1) x1 = x3^2 + x5^2
(1-x2) x2 = x4^2 + x5^2
x1 x2 = x5^2
(Those in the know will recognize Plucker coordinates; use the map
f: (a, b, c) -> (b^2, c^2, ab, ac, bc) to get a two-to-one map
from the 2-sphere into R^5.)
. . . . . . . . . . . . . . . . . . . .
But is this really "the same" as their planet? Indeed, is the sphere
really "the same" as earth? If all you want is a one-to-one
correspondence which is continuous, then yes, the earth is a
sphere. But we know the earth bulges at the equator; it has mountains
and valleys; certainly, it is not of radius 1 if we use any
conventional units.
This difficulty is noticed even by earthly ants: although they can
easily set up coordinates around their homes and make maps, these maps
fail to preserve angles and lengths. If we use latitude and longitude
to equate portions of the earth with portions of the Euclidean plane,
then paths which on the map appear to be the shortest among all those
joining two points are often _not_ the shortest as measured by walking
on the earth.
Now the ants are considering _Riemannian manifolds_. These are manifolds
in the above sense but with the additional structure that at each point
in the manifold we have not only a one-to-one correspondence with
Euclidean space, but we have decided to use that one-to-one correspondence
to determine the inner products (dot products) of vectors at the point.
In particular, we now have a notion of _lengths_ of vectors and _angles_
between vectors. (Since this is all relative to a point, computing the
length of a _curve_ requires integrating direction vectors along a curve and
so on, which makes the subject rather heavy reading.) Of course, Euclidean
space is a Riemannian manifold, and so on any submanifold one can use the
inner product of the surrounding space as the inner product on the
submanifold. So, sure, you can obviously set up a smooth one-to-one
correspondence between the points of a sphere and the points of an
ellipsoid (both of them submanifolds of R^3), but you can't do so and
hope to preserve angles and lengths; they are _the same_ as
differentiable manifolds but _different_ as Riemannian manifolds.
A mapmaker can convey all the information available in a Riemannian
manifold. For example, a Mercator projection of (a portion of) the
Earth gives a continuous one-to-one correspondence of points, and even
manages to preserve angles, but it must be equipped with a little
graph which shows a factor by which lengths must be reduced according
to latitude.
So perhaps Moriarity's question was intended to include this metric
information. Suppose we ask whether every Riemannian manifold M is
_isometric_ to a submanifold of Euclidean space. This asks if there is
a submanifold M' of R^n such that a smooth 1-to-1 correspondence
between M and M' may be set up, which also preserves lengths and angles.
Using Whitney's theorems, one can certainly find an M' in R^n and
the smooth correspondence, but there's no guarantee that the metric
information will be at all correct.
In 1954, John Nash proved that if M is a (compact) n-dimensional Riemannian
manifold, and has any embedding at all in R^N, then it also has an
isometric embedding in R^N (assuming N > n). Combined with the Whitney
theorems, this shows M is isometric to a submanifold of R^(2n).
(He also proved that non-compact manifolds can be isometrically embedded,
but I don't remember what this does to the embedding dimension.)
There is a technical problem here related to smoothness. Nash showed that
there is indeed an isometry f : M --> M' which is one-to-one and even
differentiable, but not necessarily as smooth as M. He improved this in
1956 to show that if M is endowed with a C^k metric, then there is
a C^k isometry f: M --> M' to a submanifold M' of R^N as long
as N is at least n(3n+11)/2. This result applies only for k > 2,
but it does apply to k=infinity, that is, real-analytic manifolds may be
real-analytically embedded. Non-compact manifolds may be handled as
well; we need N >= n(n+1)(3n+11)/2 in that case.
There has been considerable work since then as well, although I would
say the problem of the minimal embedding dimension is rather more
difficult than in the non-metric setting.
(Here perhaps the experts can help me: it would appear from this that
there is an isometric embedding of the _flat_ torus M = S^1 x S^1 into
R^3, using a map which cannot be expected to be twice-differentiable.
Can such an embedding be given explicitly? I don't need the Nash
theorem to find a smooth isometric embedding of M into R^17, thank you.)
All the papers above appeared in the Annals of Math, so as you may
surmise they were and are considered deep and important.
Both Whitney and Nash are fascinating characters. A great deal of
attention was focused on Nash about a year ago (e.g. articles in the
Intelligencer) when he was awarded the Nobel Memorial prize in
Economics for some game theory work he did in the 1950s. (His proofs
are deeply rooted in functional analysis, not the geometry or game
theory one might expect.)
The study of manifolds is Differential Topology; the study of
Riemannian manifolds is Differential Geometry. Each has become
incredibly robust, with many directions of research, fascinating
examples, and significant applications. The material is not easy, and
requires a good background in topology and analysis. Spivak's
"Comprehensive Introduction to Differential Geometry", a mere 3000
pages, is an excellent resource starting at the undergraduate level.
dave
==============================================================================
From: ksbrown@seanet.com (Kevin Brown)
Newsgroups: sci.math
Subject: Re: Question about non-euclidian geometry
Date: Wed, 21 Aug 1996 06:47:37 GMT
keylime@indy2.uucp (Greg Moriarty) wrote:
> Can one go from a non-euclidian geometry to a euclidian geometry by
> introducing a new dimension? Is this something that is done? If so,
> can one ALWAYS do so?
Since you specified introducing just _a_ new dimension (singular),
the answer is no. This was proved around 1901 by Hilbert, who showed
that the original non-Euclidean space (the 2D hyperbolic plane of
Lobachevski, Bolyai, et al) cannot be embedded in its entirety in
3D Euclidean space. However, it CAN be embedded in 4D Euclidean
space. As others have mentioned, you can always embed a smooth
metrical non-Euclidean space in a higher-dimensional Euclidean
space, but it usually takes more than just one extra dimension.
> I would think such a question would have practial implications. For
> example, Minkowski introduced a 4th dimension to give SR a geometric
> description. Einstein went a step further by making that 4-D space
> non-euclidean. Could one introduce a 5th dimension to return to a
> Euclidean universe?
The reference to Minkowski complicates the question a bit. If
you're willing to allow imaginary "distances" (i.e., distances whose
squares are negative), then it IS possible to embed the hyperbolic
plane in 3D Euclidean space - as the surface of a sphere with
imaginary radius. In fact, as early as 1766 Lambert observed that
if you assumed there are at least two distinct lines through a given
point that don't intersect a given line, then the area of a triangle
with angles a,b,c would be -R^2 (a+b+c-pi) for some constant R. He
knew that the area of a triangle on a real sphere of radius R was
R^2 (a+b+c-pi), so he wrote "one could almost conclude that the new
geometry would be true on a sphere of imaginary radius". It turns
out that if you just plug in iR as the radius in any formula for
spherical geometry you get the corresponding formula for hyperbolic
geometry.
On the other hand, it isn't clear that a formally Euclidean space
with imaginary distances is any more intuitive than a curved space
with strictly real distances. In fact, I would argue that the most
un-intuitive aspect of the relativistic "metric" of spacetime is not
it's curvature, but the fact that it isn't really a metric at all.
A metric space, in the strict sense of the term, is a manifold that
satisfies the triangle inequality, which is the property that leads
to our intuitive impressions of "locality". In particular, locality
is transitive in a metric space, meaning that if A is close to B,
and B is close to C, then A can't be too far from C. Spacetime
doesn't satisfy this condition. There exist points A,B,C such that
the absolute distances AB and BC are both less than the Planck length
(10^-35 meters), and yet the distance AC is the radius of the
observeable universe. This has nothing to do with curvature; it's
strictly a consequence of the fact that spacetime is not a metrical
space and does not satisfy the triangle inequality. In view of this,
a little bit of curvature should be the least of your worries.
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| MathPages / \ http://www.seanet.com/~ksbrown/ |
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I'd like to clarify a few issues of my post above.
First let me acknowledge James Dolan's objection to my one-line
summary of differential geometry. Surely historical precedent dictates
that the right response be, "Differential geometry is what differential
geometers do"; equivalently it's just AMS Subject Classification 53.
My Columbia Encyclopedia (with Hyman Bass overseeing Mathematics entries)
gives it as the "branch of geometry in which the concepts of the
calculus are applied to curves, surfaces, and other geometric entities".
Second, I'd like to describe better the planet on which the second set
of ants live -- the projective plane RP^2. As I pointed out, this surface
cannot be embedded in R^3, but it can be _immersed_ in R^3. (Immersions
are maps which are locally homeomorphisms but which allow for
self-intersections -- the standard picture of a Klein bottle is an
immersion.) The classic immersion is the so-called Boy's surface.
You can find some pretty pictures and parametric equations at the
Geometry Center,
URL: http://www.geom.umn.edu/zoo/toptype/pplane/
The map (f,g,h) describing the Boy's surface there is one-to-one except along
a curve of double points; thus the map (f,g,h,k) : S^2 --> R^4 gives
an embedding of the projective plane into R^4 as long as k is a
homogeneous polynomial of even degree which separates collapsed points
along that curve.
Third, please ignore all references to real-analyticity in my previous
post. Nash did considerable work in the analytic and algebraic categories,
but I'm not qualified to say what it was. I must have been very tired
when I confused C^\infinity ("smooth") with C^\omega ("analytic").
Finally, I still seek a description of a flat submanifold of R^3 which
is diffeomorphic to the flat torus.
==============================================================================
EQUATIONS FOR BOY'S SURFACE
Let
F = ( f(x, y, z), g(x, y, z), h(x, y, z) )
be a map from R^3 to R^3. If the functions
f, g, h
are homogeneous polynomials of even degree >= 2, then for all (x, y,
z) in R^3 we have
F(x, y, z) = F(-(x, y, z)).
Thus the restiction of F to the unit 2-sphere sends antipodal points
to the same image point--hence, F defines a new map from RP^2 to R^3.
In the case of Boy's surface,
f(x, y, z) = [ (2x^2 - y^2 - z^2)(x^2 + y^2 + x^2) + 2yz(y^2 -
z^2) + zx(x^2 - z^2) + xy(y^2 - x^2) ]/2
g(x, y, z) = (Sqrt(3))/2 [ (y^2 - z^2)(x^2 + y^2 + z^2) + zx(z^2 -
x^2) + xy(y^2 - x^2) ]
h(x, y, z) = (x + y + z)[ (x + y + z)^3 + 4(y - x)(z - y)(x -
z)]
Pictures may be created parametrically in Maple by the following
commands:
x:=cos(t)*sin(s);
y:=sin(t)*sin(s);
z:=cos(s);
f:=1/2*((2*x^2-y^2-z^2) + 2*y*z*(y^2-z^2) + z*x*(x^2-z^2)
+x*y*(y^2-x^2));
g:= sqrt(3)/2*((y^2-z^2) + z*x*(z^2-x^2) + x*y*(y^2-x^2));
h:=(x+y+z)*((x+y+z)^3 + 4*(y-x)*(z-y)*(x-z));
plot3d( [h/8,f,g], s=0..Pi, t=0..Pi)
_________________________________________________________________
Up: Boy's Surface
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