Circles and Clifford Algebras
"
V. Timorin
Abstract. We study smooth maps from an open subset of a real projective space to
a sphere such that all germs of straight lines go to germs of circles (or points). We give
a construction of such maps based on representations of Clifford algebras. It provides
a complete purely geometric description of Clifford algebras representations in terms of
circles. We also describe a connection between our problem and the Hurwitz Radon
theorem about sums of squares.
1 Introduction
By a circle in a Euclidean space we mean an honest Euclidean circle, or a straight
line, or a point. Our problem is as follows: describe all smooth (or continuous)
maps from an open subset of Rm (or a projective space RPm) to a Euclidean space
Rn (or a sphere Sn) that take germs of straight lines to germs of circles. Orig-
inally, this problem came from nomography and was stated for diffeomorphisms
between open subsets of R2 by G. S. Khovanskii. Nomography is about graphical
representations of functions. Nomograms using a compass are more accurate than
those using a ruler. Thus the above problem arises. The original 2-dimensional
problem was solved by A. Khovanskii in [1]. He proved that, up to a projective
transformation in the space of pre-image and a Mbius transformation in the space
of image, there are only 3 smooth one-to-one maps taking lines to circles. They
correspond to the 3 classical geometries Euclidean, spherical and hyperbolic.
The Euclidean map is just the identity. The spherical map is the composition of
a central projection to a sphere and a stereographic projection back to the plane.
The hyperbolic map is the same with a hyperboloid instead of a sphere. This is
the map between the Klein model of hyperbolic geometry and the Poincar unit
disk model.
"
Partially supported by CRDF RM1-2086
1
arXiv:math.MG/0210212 v1 15 Oct 2002
In a unified way all the maps from Khovanskii s theorem may be described as
follows. To get a map from an open subset of R2 to a sphere, one should first
embed R2 to R3 as an affine plane and then project it to a sphere from some point.
If we project from inside of the sphere, then we get the spherical case. If the
center of projection lies on the sphere, then we have the Euclidean case. Finally,
projecting from outside of the sphere, we get the hyperbolic case.
F. Izadi in his PhD thesis [2] carried over the results of Khovanskii to the
3-dimensional case.
But in dimension 4 this is wrong! There are more maps taking lines to circles.
For example, identify R4 with C2 and consider a complex projective transforma-
tion. It takes complex lines to complex lines, and on each complex line it acts as
a Mbius transformation. Note that every real line belongs to a unique complex
line. Thus any real line goes to a circle.
The author in [3] constructed a large class of maps from an open subset of R4 to
R4 that take lines to circles. Let A and B be 2 affine maps from R4 to R4. Identify
R4 with the algebra H of quaternions. Consider the maps x A(x)B(x)-1 and
x B(x)-1A(x). Here the multiplication and the inversion are in the sense of
quaternions. These maps are called fractional quaternionic transformations. They
take lines to circles. Note that fractional quaternionic transformations are not
necessarily fractional linear: A and B are affine over reals, not over quaternions.
Fractional quaternionic transformations do not give a complete solution to the
problem. There are other maps taking lines to circles (e.g. the standard spherical
and hyperbolic maps). Thus the problem remains open in dimension 4.
In [3], the following micro-local theorem is proved. Consider a germ of dif-
feomorphism Ś : (R4, 0) (R4, 0) that takes germs of lines passing through 0 to
germs of circles. Then the set of circles in the image is the same as for a frac-
tional quaternionic transformation preserving 0. This describes so called rectifiable
bundles of circles (sets of circles passing through some point and obtained from
straight lines via some local diffeomorphism). In dimensions 2 and 3, the situation
is much simpler: any sufficiently large rectifiable bundle of circles at 0 has some
other common point of intersection, i.e., it is obtained from a bundle of lines by
some Mbius transformation [1, 2].
In this paper, we study the multidimensional case. We give a construction
of continuous maps from RPm to Sn that take all lines to circles or points. This
construction is based on representations of Clifford algebras Cliff(r) generated over
R by r anti-commuting imaginary units. With any representation Ć of Cliff(r) in
Rn we associate a map from an open subset of Rr+n+1 to Rn that takes lines to
circles. Namely, Rr+1 is embedded into Cliff(r) as the subspace spanned by all
2
generators and 1. A map FĆ from (Rr+1 - 0) Rn to Rn sends a pair (ą, x) to
Ć(ą)-1(x). This map takes lines to circles. Projectivizing the space of preimage
and compactifying the space of image we obtain a map from RPn+r to Sn with the
same property.
We will prove the following micro-local theorem. Consider a germ of a smooth
map Ś : (RrRn, 0) (Rn, 0) that takes germs of lines passing through 0 to germs
of circles. Assume that Ś(ą, x) equals to x plus higher order terms vanishing
whenever ą = 0. Then the set of circles in the image is the same as for some map
FĆ or its restriction to a hyperplane.
The maps FĆ : Rm Rn give many examples of rectifiable bundles of circles.
It is enough to compose FĆ with a generic affine embedding Rn Rm to get a
local diffeomorphism sending lines to circles.
Let us return to dimension 4. The algebra Cliff(3) is isomorphic to H " H.
Thus it has 2 representations in R4 = H: by the left quaternionic multiplications
and by the right quaternionic multiplications. This gives 2 maps from R8 to R4
taking all lines to circles. Note that all circles are images of lines under these maps.
All fractional quaternionic transformations may be obtained from these maps as
described above. We also get 2 maps from RP7 to S4. These are nothing more
than Hopf fibrations! It is a remarkable fact that they take all lines to circles.
The paper is organized as follows. In Section 2, we recall some basic facts about
Clifford algebras and their representations. Section 3 contains the construction of
maps FĆ and a proof that these maps take lines to circles. Then we formulate the
micro-local theorem in Section 4. Section 5 establishes a connection between this
theorem and the famous Hurwitz Radon theorem about sums of squares. Finally,
in Section 6 we conclude the proof of the micro-local theorem and give a description
of Clifford algebras representations as linear spaces of complex multiplications.
I am grateful to A. Khovanskii for useful discussions.
2 Clifford algebras
Let V be a real vector space equipped with a symmetric bilinear form Q. The
Clifford algebra Cliff(V, Q) is the associative algebra with a unit generated by all
vectors from V subject to relations vw + wv = Q(v, w) for all v, w " V . If V
has dimension r and Q is negative definite, then the algebra Cliff(V, Q) is denoted
by Cliff(r). In other words, Cliff(r) is generated by r anti-commuting imaginary
units e1, . . . , er:
e2 = -1, eiej + ejei = 0 (i = j).
i
3
We will be always dealing with Clifford algebras of type Cliff(r). For a Euclidean
space V with the Euclidean form q we denote the algebra Cliff(V, -q) by Cliff(V ).
A Clifford algebra is a direct generalization of the skew-field of quaternions H =
Cliff(2).
The following table gives the Clifford algebras Cliff(r) for r < 8.
r 0 1 2 3 4 5 6 7
Cliff(r) R C H H " H H[2] C[4] R[8] R[8] " R[8]
Here A[r] (A = R, C or H) denotes the algebra of r r-matrices over A. The
other Clifford algebras are computed by Bott periodicity
Cliff(n + 8) = Cliff(n) " R[16].
This table was first obtained by E. Cartan [4], see also [5]. All real representations
of Clifford algebras read from the table since a matrix algebra A[r] has a unique
irreducible representation, namely, Ar, and all representations are completely re-
ducible.
Example 1. The algebra Cliff(3) is isomorphic to H"H. In this presentation,
e1 = (i, -i), e2 = (j, -j) and e3 = (k, -k).
Example 2. The algebra Cliff(7) acts on R8 = H2. Let us describe the action
of the space generated by 1 and ei (i = 1, . . . , 8). The latter is also identified with
H2. For any quaternion a denote by La the operator of left multiplication by a in
H and by Ra the operator of right multiplication. A pair (a, b) " H2 acts as
La -Rb
Ż
.
Rb L
If we interchange the left and right multiplications, then we get the other repre-
sentation of Cliff(7) in R8.
Ć
Consider a Euclidean vector space V . Denote by V the space V " 1 which lies
in Cliff(V ) and is generated by V and 1. Let us say that a linear representation of
Cliff(V ) in Rn is compatible with the Euclidean metric (or with Euclidean inner
Ć
product) if any vector from V acts conformally, i.e. multiplies all distances by the
same factor. This restriction is not on the representation but on the Euclidean
metric on Rn only:
Proposition 2.1 For any linear representation of Cliff(r) in Rn there exists a
Euclidean metric on Rn such that the representation is compatible with it.
4
This fact is well-known but we sketch a proof here. Consider the Dirac group
consisting of all products of generators e1, . . . , er. Obviously, the Dirac group is
finite. Therefore, it has an invariant Euclidean metric. In particular, all elements
ei act as orthogonal operators with respect to this metric. A direct computation
now shows that all linear combinations of 1 and the generators act conformally.
A representation of a Clifford algebra in a Euclidean space is always assumed
to be compatible with the Euclidean inner product.
Linear representations of the algebras C = Cliff(1) and H = Cliff(2) in Rn are
called complex and quaternionic structures on Rn, respectively. The image of i
under a representation of C is also called a complex structure. This may be any
linear operator I such that I2 = -1. By a complex (quaternionic) structure on
a Euclidean space we always mean a complex (quaternionic) structure compatible
with the Euclidean metric. For a complex structure I this means that I is orthog-
onal and skew-symmetric. For a quaternionic structure, from a compatibility it
follows that all quaternions act conformally, not only linear combinations of 1, i
and j.
3 Main construction
We need the following very simple (and well-known) lemma:
Lemma 3.1 Consider the following fractional linear map from A to Ak (where A
is C or H):
F : x (xc + d)-1(xa + b).
Here x, c, d " A and a, b " Ak. Assume that the target space is equipped with a
Euclidean metric compatible with the complex or quaternionic structure. Then F
takes all real lines to circles.
Proof. We have F (x) = A + (xc + d)-1B where A = c-1a and B = b - dc-1a.
Therefore the image of A lies on the left line {A + uB| u " A}. The coordinate
u on this line is linear and conformal. The map x u = (xc + d)-1 is a Mbius
transformation. Hence any real line goes to a circle.
Here is the main construction. Let V be a Euclidean space and Ć a represen-
Ć
tation of Cliff(V ) in Rn. Recall that V = V " 1 " Cliff(r).
Ć
Theorem 3.2 Consider the map FĆ : (V - 0) Rn Rn given by (ą, x)
Ć-1(ą)x. This map takes all lines to circles.
5
Ć
Proof. Consider a line L in V Rn. If it passes through 0, then its image
is empty or a single point. Suppose now that L does not contain the origin. The
Ć
projection of L to V generates a subalgebra of Cliff(V ) which is isomorphic either
to C or to H.
In the first case L lies in C Rn. The action of C defines a complex structure
on Rn. Thus Rn may be identified with Cn/2. Let t be a linear parameter on L.
Then the image of L in Rn is given by ą(t)-1x(t) where x(t) is a vector from Rn
and ą(t) is a complex number, both depend affinely on t. Now from Lemma 3.1 it
follows that the image of L is a circle.
The quaternionic case is completely analogous.
Remark. The above construction may be slightly improved. Note that under
the map FĆ from Theorem 3.2 proportional vectors have the same image. There-
Ć
fore, there is a map from the projectivization of V Rn to Sn (the latter is obtained
from Rn by a one-point compactification). This map takes real projective lines to
circles. The projectivization of 0 Rn goes to infinity. The map is continuous
everywhere but may be non-smooth at the preimage of infinity. Our construction
gives the following maps:
RP3 S2, RP7 S4, RP7 S6, RP15 S8, . . .
In general, we get a map from RPn+(n)-1 to Sn where (n) is the Hurwitz-Radon
function which is the largest r for which there is a representation of Cliff(r - 1) in
Rn. It may be also defined as follows. Let n = 2su where u is odd. If s = 4a + b,
0 b 3, then (n) = 8a + 2b.
The maps above seem to be known to algebraic topologists but they were
considered only in the homotopy theoretic context.
4 Micro-local theorem
Our main purpose is to prove the following theorem:
Theorem 4.1 Consider a germ of a smooth map Ś : (Rr Rn, 0) (Rn, 0) that
takes germs of lines at 0 to germs of circles. Assume that Ś(ą, x) equals to x plus
higher order terms vanishing when ą = 0. Then there is a Euclidean space V , an
Ć
embedding of Rr into V = V " 1 " Cliff(V ) as a vector subspace of codimension
at most 1 and a representation Ć of Cliff(V ) in Rn such that the map
F : Rr Rn Rn, (ą, x) Ć(1 - ą)-1x
gives the same family of circles in the image as Ś does. In particular, r (n).
6
Theorem 4.1 gives a geometric description of representations of Clifford algebras
in terms of circles.
Remark. The maps F and Ś do not need to be the same. For instance, if we
first multiply everything by a smooth function and then apply F , then we get the
same family of circles in the image.
The map Ś from theorem 4.1 has the form
Ś(ą, x) = x + (ą, x) + "(ą) + . . . ,
where is some bilinear map Rr Rn Rn, the form " is quadratic on Rr, and
dots denote higher order terms.
Lemma 4.2 The form " vanishes for all ą. For any fixed ą the quadratic forms
(, ) and (x, ) are divisible by (x, x).
The proof of this lemma is the same as for Proposition 1.3 in [3]. We proved
there that the second differential at 0 of a local diffeomorphism Ś : (Rn, 0)
(Rn, 0) taking lines through 0 to circles must satisfy the above divisibility condi-
tions provided that the first differential is the identity. But we never used that Ś
was a diffeomorphism. So the proof applies equally well to our smooth map Ś.
It is worth repeating here the informal reason for Lemma 4.2 to hold. Assume
that Ś is analytic. Then Ś admits an analytic continuation to a neighborhood of
0 in Cn+r. Denote by C the cone {(x, x) = 0} in Cn where (, ) is the analytic
continuation of the Euclidean inner product. The analytic continuation of Ś takes
germs of complex lines to germs of complex circles , i.e., plane algebraic curves
of second degree asymptotic to C.
Denote by Ą the natural projection from RrRn to Rn and its complexification,
a projection form Cr Cn to Cn. Take a line L from Ą-1(C). It goes to a complex
circle. Suppose that this circle is not a line. Then the whole plane containing the
circle belongs to C. Indeed, in the opposite case the intersection of this plane with
C would be a pair of intersecting lines (possibly coincident). The circle passes
through a point of their intersection and is asymptotic to one or both of them.
But this is impossible for a second-degree curve.
Take a vector y from L. The image of L is a complex circle tangent to x = Ą(y).
We know that either this circle coincides with Ą(L) or the whole plane containing
it belongs to C. In both cases the linear span of x and Ś(y) belongs entirely to C.
In particular, the functions (x, Ś(y)) and (Ś(y), Ś(y)) vanish on C. Hence they
are divisible by (x, x) (as analytic functions of y). The Lemma 4.2 now follows.
7
Note that the bilinear map contains all information about circles in the
image of Ś. Indeed, to define a circle it is enough to know its velocity and
acceleration at 0. Thus theorem 4.1 reduces to a description of the form .
5 Connection with the Hurwitz Radon theorem
We are going to explore the properties of the bilinear map : Rr Rn Rn
introduced in the previous section. Recall that Rn is equipped with a Euclidean
metric. Up to now, Rr is only a vector space. But we will introduce a Euclidean
metric on it as well. Fix an element ą " Rn. It gives rise to a linear map
A : Rn Rn given by A(x) = (ą, x). By Lemma 4.2 this map is conformal.
This means that the product of A with its conjugate is the multiplication by a real
number q(ą). It is readily seen that q is a positive definite quadratic form on Rr.
Thus Rr may be regarded as a Euclidean space.
Now denote by | | the Euclidean norms in both Rr and Rn. Then we have
|(ą, x)| = |ą| |x|.
Introduce orthonormal bases in Rr and Rn. Denote the coordinates of x by
x1, . . . , xn, the coordinates of ą by ą1, . . . , ąr and the components of by
ł1, . . . , łn. Then we have
2 2 2 2
(ą1 + + ąr)(x2 + + x2) = ł1 + + łn.
1 n
Hurwitz [6] and Radon [7] described all relations of this form. In terms of Clifford
algebra representations their result can be formulated as follows [5].
Theorem 5.1 Let f : Rk+1 Rn Rn be a bilinear map such that |f(ą, x)| =
|ą| |x| for all ą " Rk+1 and x " Rn. Then there is a representation Ć of the
Clifford algebra Cliff(k) in Rn such that
f(ą, x) = Ć(ą0 + ą1e1 + + ąkek)A0(x)
where A0 is a linear conformal transformation and ą0, . . . , ąk are coordinates of ą
in some orthonormal basis.
In particular, this theorem may be applied to our bilinear map . But we know
more about and hence we can say more. Theorem 5.1 has a degree of freedom:
an arbitrary linear conformal transformation in the preimage. Our additional
restriction on kills this degree of freedom.
Some interesting generalizations of theorem 5.1 and the general Hurwitz prob-
lem are discussed in [8].
8
6 Complex multiplications and Clifford algebras
In this section, we establish a connection between the bilinear map and a rep-
resentation of some Clifford algebra. This connection is in the spirit of Theorem
5.1, and we use very similar arguments.
Fix a Euclidean metric on Rn. The inner product of 2 vectors x, y " Rn is
denoted by (x, y). A linear operator A : Rn Rn is called a complex multiplica-
tion if either A is the multiplication by a real number, or there exists a complex
structure I on Rn compatible with (, ) such that A is the multiplication by a
complex number with respect to I.
Here is a characterization of all complex multiplications in real terms.
Proposition 6.1 A linear operator A : Rn Rn is a complex multiplication if
and only if the quadratic forms (x, Ax) and (Ax, Ax) are both divisible by (x, x).
Proof. If A = a + bI for some complex structure I, then (x, Ax) = a(x, x)
and (Ax, Ax) = (a2 + b2)(x, x).
Now assume that (x, Ax) = p(x, x) and (Ax, Ax) = q(x, x) for some real num-
bers p and q. From the Cauchy-Schwartz inequality it follows that q e" p2. If
q = p2, then A is the multiplication by p. If q > p2, then A = p + q2 - p2I where
I is a linear operator that is orthogonal and skew-symmetric. The orthogonality
means that I"I = 1 where I" is the conjugate operator to I. The skew-symmetry
says that I" = -I. Hence I2 = -1, i.e. I is a complex structure, and A is a
complex multiplication.
Let : Rr Rn Rn be the bilinear map introduced in Section 4. Identify
Rr with a space of complex multiplications: a vector ą " Rr gets identified with
the operator x (ą, x) which is a complex multiplication by Lemma 4.2 and
Ć
Proposition 6.1. Denote by V the linear span of Rr and the identity. There is
Ć
a Euclidean form on V . Namely, for each complex multiplication A put q(A) =
(A(x), A(x))/(x, x) which is independent of x. Clearly, q is a positive definite
quadratic form.
Ć
Proposition 6.2 Let V be the orthogonal complement to 1 in V with respect to
q. Then there exists a representation Ć of Cliff(V ) in Rn such that for any ą " Rr
Ć
and x " Rn we have (ą, x) = Ć(ą)(x) under the natural embedding of V into
Cliff(V ).
Proof. Denote by q(, ) the polarization of the quadratic form q(). By
Ć
definition, for any A " V and x " Rn we have (Ax, Ax) = q(A)(x, x). Therefore
9
Ć
any 2 elements A, B " V satisfy (Ax, By) + (Bx, Ay) = 2q(A, B)(x, y) identically
with respect to x, y " Rn. In other words, A"B + B"A = 2q(A, B).
Ć
Choose an orthonormal basis in V containing 1. Denote the other elements of
this basis by I1, . . . , Ir. We have
" 2 "
Ij + Ij = 2q(1, Ij) = 0, Ij = -Ij Ij = -q(Ij) = -1,
" "
IjIk + IkIj = -(Ij Ik + IkIj) = -q(Ij, Ik) = 0.
Thus the operators I1, . . . , Ir give rise to a representation of Cliff(V ).
Proof of Theorem 4.1. By Proposition 6.2, the bilinear map sends (ą, x)
Ć
to Ć(ą)(x) for some embedding Rr V and some representation Ć of Cliff(V ) in
Rn. Thus the 2-jet of Ś at 0 is the same as that of F : (ą, x) Ć(1 - ą)-1x. But
the circles in the image are determined by the 2-jet only. Hence under the maps
Ś and F the same lines get mapped to the same circles.
References
[1] A.G. Khovanskii: Rectification of circles, Sib. Mat. Zh., 21 (1980), 221 226
[2] F.A. Izadi: Rectification of circles, spheres, and classical geometries, PhD the-
sis, University of Toronto, (2001)
[3] V.A. Timorin: Rectification of circles and quaternions , Preprint
http://xxx.lanl.gov/math.DG/abs/0110144. To appear in Michigan Mathemati-
cal Journal
[4] E. Cartan: Nombres complexes, pp. 329-448 in J. Molk (red.): Encyclopdie
des sciences mathmatiques, Tome I, Vol. 1, Fasc. 4, art. 15 (1908). Reprinted
in E. Cartan: OEuvres complŁtes, Partie II. Gauthier-Villars, Paris, 1953, pp.
107-246
[5] M.F. Atiyah, R. Bott, A. Shapiro: Clifford modules, Topology 3, suppl. 1
(1964), 3-38. Reprinted in R. Bott: Lectures on K(X). Benjamin, New York,
1969, pp. 143-178. Reprinted in M. Atiyah: Collected Works, Vol. 2. Clarendon
Press, Oxford, 1988, pp. 301-336
[6] A. Hurwitz: ber die Komposition der quadratischen Formen, Math. Ann. 88
(1923) 1-25. Reprinted in Math. Werke II, 641-666
10
[7] J. Radon: Lineare Scharen orthogonaler Matrizen, Abh. Math. Sem. Univ.
Hamburg 1 (1922), 1-14
[8] D. B. Shapiro: Compositions of Quadratic Forms, de Gruyter Expositions in
Math., 33 (2000)
11
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