xu,
F = xu
xv, G = xv xv,Premise. For 300 years or more, all sorts of ways have been invented to construct surfaces in Euclidean 3-space. But surfaces in 4-space have been largely neglected--they seem to be too advanced for elementary books and too elementary for advanced books. The goal here is to list some well known ones--not only in R4 but also in higher dimensions.
Computers. Gaussian curvature K is the most important geometric invariant of a surface (more about K below). Thus to be interesting, a surface ought to have reasonable Gaussian curvature. For surfaces in 3-space there is a standard formula that is not hard to compute by hand. But for higher dimensions, hand computation of formulas for K can get nasty, so we will describe later how to download computer commands for K from:
Mathematica or Maple.
These references outline the general syntax of the two systems and the computer tools that are most useful in studying curves and surfaces. For anyone interested in differential geometry, these pages may be a good introduction to Mathematica or Maple.
Help! If you notice any errors, please report them to bon@math.ucla.edu.
Surfaces. Surfaces are very intuitive objects. We usually describe them parametrically, that is, as the image of a regular mapping x from a region D in R2 into Rn. Typically the coordinates of R2 will be u,v.
The precise way that x distorts the flat Euclidean plane R2 is measured by its metric components,.
xu,
F = xu
xv, G = xv xv,where xu is the vector derivative of x with
respect to u, and () is the dot
product in Rn.
In the extreme case E=1, F=0, G=1, the vectors
xu and xv are orthonormal, so
there is no distortion. And if x is also one-to-one, it is a
simple example of an isometry.
Note. The term "regular" applied above to a
mapping x means that at each point the vectors
xu and xv are linearly independent,
guaranteeing that, locally at least, the image of x really is a
surface.
Curvature. The
Gaussian curvature K of a surface is a
real-valued function that in the case of a surface in
R3 locally describes its shape:
K(p) > 0 means
shaped like an ellipsoid near p,
K(p) < 0 means
saddle-shaped.
K = 0 isn't decisive
unless it holds over a whole region, which is then said to be
flat (though "flattenable" might be more descriptive).
Usually, K will be given as a function on D with K(u,v) the
curvature of the surface at the point x(u,v).
For surfaces in higher dimensions, curvature is still
crucial, but its geometric meaning is not so obvious.
The Mathematica page
contains the following three computer commands for curvature;
the Maple page contains only the first (and most useful):
"gaussK" Fastest, but
applies only to surfaces in R3.
"tensorK" Computes K for
surfaces whose metric coefficients E, F, G are given.
"superK" An extension of tensorK that gives K for surfaces in any Rn, n
3.
Near these commands are instructions for downloading to your computer.
1. A Flat Torus in R4.
2. Small
Euclidean Planes in R4.
3. Graphs of Functions in R4.
4. A Non-round Sphere in R4.
5. A Flat Klein Bottle in R5.
6. A Projective Plane P in R5.
7. The
Hyperbolic Plane in R6.
1. A Flat Torus in R4
The usual surface-of-a-doughnut torus of revolution in R3 is certainly curved. But in R4 there is a more natural model that is flat (i.e., has curvature K=0). If C is the unit circle in R2, then taking Cartesian products gives
torus = C × C in R2 × R2 = R4
To verify computationally that this torus is flat, use the parametrization
flattorus(u,v) = (cos u, sin u, cos v, sin v).
Now xu = (-sin u, cos u, 0, 0), and xv = (0, 0, -sin v, cos v).
Evidently xu and xv are orthonormal, that is, E=1, F=0, G=1, so this torus is flat.
2. Small Euclidean Planes in R4
These are exact (i.e., isometric) one-to-one copies of the
Euclidean plane R2, but they can be found inside
a ball of arbitrarily small radius 
> 0 in R4.
To get one, just replace the circle C above by a spiral, say,
spiral(t) = (2+tanh t) (cos t, sin t)
This spiral is a smooth one-to-one map that approaches
r=1 as t 
-
and r=3 as t + <"mc/infty.gif">.
By elementary calculus, it can be reparametrized to have to
have unit speed
on the whole real line since (as is easily checked) the speed
of spiral is bounded above and away from 0.
If t (f(t),g(t)) is the unit-speed
spiral, then for any > 0, an -small plane is given by
smallplane(u,v) =
(f(u/), g(u/), f(v/), g(v/))
Evidently this map is one-to-one, and since it has E=1, F=0, G=1, it doesn't distort R2 at all. The inhabitants of R4 see it as infinitely crumpled, but the inhabitants of the plane itself cannot tell it from R2.
3. Graphs of Functions
In R3, a natural source of surfaces is as
graphs of real valued functions f: D
R on regions D in the plane. This works in general for
differentiable functions F: D
Rn into any Euclidean space. If
F=(f1,...,fn), define
mongeF(u,v) = (u, v, f1(u,v),..., fn(u,v)).
To get a surface in R4, a natural choice
is to use a complex function C C.
Let's try the case
zzn.
In polar form--with 
given as "q"
for the computer, the graph of zn is
parametrized by
(Mathematica):
polarzn[r_,q_] = {r*Cos[q], r*Sin[q], r^n*Cos[n*q], r^n*Sin[n*q]}
(Maple): polarzn := (r,q) -> [r*cos(q), r*sin(q), r^n*cos(n*q), r^n*sin(n*q)]
SuperK shows that (for n>1) the curvature K of this
surface is always negative.
This is true in general:
Lemma. The Gaussian curvature K of the graph of a (nonlinear) complex analytic function f+ig is always negative. Explicitly,
K = -2(fuu2+ fuv2) / (1+fu2+ fv2)3.
4. A Non-round Sphere in R4
A standard "round" sphere S of radius r in R3 is given by ||x||=r. It has constant positive curvature K=1/r2 as can be computed from the parametrization
x(u,v) = r (cos v sin u, cos v cos u sin v, sin v),
expressed in terms of either Mathematica or Maple.
A theorem of Liebman shows that the sphere is rigid in R3, that is, any surface with the same intrinsic geometry (i.e., a topological sphere with constant K) must also be round. By contrast, a plane in R3 can easily be given troughs or ripples without changing its intrinsic geometry.
However, the rigidity of the sphere fails in R4. This can be shown by the following example: a topological sphere in R4 that has constant positive curvature K, but is not round (which means, in particular, that it is not contained in any 3-dimensional hyperplane in R4).
5. A Flat Klein Bottle in R5
Abstractly, a flat Klein bottle B can be constructed from a square in the (flat) Euclidean plane, as suggested in the figure, by sewing the top and bottom sides together to form a piece of pipe, then sewing the ends together "backwards" to give the Klein bottle. (Connecting the ends in the ordinary way would give a torus.)
Any compact surface in R3 has some
positive Gaussian curvature--say, at a point of the surface
farthest from the origin. Thus B cannot be found
R3.
The familiar picture of a Klein bottle as a piece of
laboratory glassware in 3-space is certainly not flat--
furthermore, since it cuts across itself, it is only
immersed in R3, not imbedded.
An example due to C. Tompkins gives a Klein bottle in R4 that is flat, but it still cuts across itself. However, a slight modification gives a genuine imbedding -- but in one higher dimension.
6. A Projective Plane P in R5
As a geometric surface, P is gotten by identifying
antipodal points, p and -p, of a standard round sphere S
(that is, consider their fusion as a single point of P). So if S
has radius r, hence K=1/r2, the same is true of P. This describes P intrinsically, and the problem now is to isometrically imbed it in a Euclidean space.
Here is an imbedding of P in
R5. This is the same as an imbedding of the
sphere for which antipodal points are sent to the same point in
R5.
7. The Hyperbolic Plane in R6
The hyperbolic plane H is described intrinsically in the
Mathematica page and the Maple page. It's the constant
K<0 analogue of the ordinary (constant K>0) sphere
S2.
The sphere lives naturally in R3, but a
theorem of Hilbert says H can't be found there.
Here is an ingenious way to imbed H in the Euclidean space R6 of dimension 6.