Infinitesimal aspects of the Laplace operator

In the context of synthetic differential geometry, we study the Laplace operator an a Riemannian manifold. The main new aspect is a neighbourhood of the diagonal, smaller than the second neighbourhood usually required as support for second order differential operators. The new neighbourhood has the property that a function is affine on it if and only if it is harmonic.


Introduction
Recall [3], [4], [5], [7] that any manifold M, when seen in a model of Synthetic Differential Geometry (SDG), carries a reflexive symmetric relation ∼ k (k = 0, 1, 2, ...), where x ∼ k y reads "x and y are k-neighbours"; x ∼ 0 y means x = y; x ∼ k y implies x ∼ k+1 y. Also, x ∼ k y, y ∼ l z implies x ∼ k+l z. The set of (x, y) ∈ M × M with x ∼ k y is denoted M (k) , the "k'th neighbourhood of the diagonal", and for fixed x, the set {y ∈ M | y ∼ k x} is denoted M k (x) ("the k-monad around x"). In R n , M k (0) is denoted D k (n). Its elements u are characterized by the condition that any homogeneous polynomial of degree k + 1 vansihes on u. -In this context, a Riemannian metric on M can be given in terms of a map g : M (2) → R with g(x, x) = 0, and with g(x, y) = g(y, x) to be thought of as the "squaredistance between x and y", se [5], [6]. (Also g should be positive-definite, in a certain sense.) Given a Riemannian metric g on M, in this sense, one can construct the Levi-Civita connection [5], volume form [6], and hence also a notion of divergence of a vector field. And to a function f : M → R, one can construct its gradient vector field, and hence one can construct the Laplacian ∆ by ∆(f ) = div(grad(f )). This is what we shall not do here, rather, we shall exploit the richness of synthetic language to give a more economic and more geometric construction of ∆. The construction is more economic in the sense that the definition of ∆f (x) only depends on knowing f on a certain subset M L (x) ⊆ M 2 (x), where M 2 (x) is what is required to make the usual div grad construction work, or for defining the individual terms in the formula ∆f (x) = ∂ 2 f /∂x 2 i (x). The description of M L (x) ⊆ M, or equivalently, the description of the L-neighbour relation ∼ L , is coordinate free, see Definition 1 below, and therefore, too, is the description of ∆f and of the notion of harmonic function. We get a characterization of harmonic functions, in terms of an average-value property, which is infinitesimal in character and does not involve integration, see Theorem 1 and Proposition 8.
In Section 3 we prove that diffeomorphisms which preserve the L-neighbour relation are precisely the conformal ones. Section 4 deals with the special case of the complex plane, and Section 5 explains the "support of the Laplacian" in systematic algebraic terms.

Preliminaries
Although the notions we use are introduced in a coordinate free way, we have no intention of avoiding use of coordinates as a tool of proof. This Section contains in fact mainly certain coordinate calculations, which we believe will be useful also in other contexts where Riemannian geometry is treated in the present synthetic manner.
Working in coordinates in M means that we are identifying (an open subset of) M with (an open subset of) R n ; for simplicity, we talk about thse open subsets as if they were all of M and R n , respectively; all our considerations are anyway only local. The Riemannian metric g on M then becomes identified with a Riemannian metric on R n , likewise denoted g, and it may be written (for x ∼ 2 y) in the form of a matrix product, where x − y ∈ R n is viewed as a column matrix, and G(x), for each x, is a symmetric positive definite n × n matrix.
Using coordinates, we may form affine combinations of (the coordinate sets of) points of M, at least for sufficiently nearby points, and such combinations will in general have only little geometric significance, since they depend on the choice of the coordinate system. However, we have the following useful fact: in particular, for x = 0, g(0, y 2 − y 1 ) = g(y 1 , y 2 ).
Proof. We may assume x = 0. Then g(0, y 2 − y 1 ) and g(y 1 , y 2 ) are given, respectively, by where H depends linearily on y ∼ 1 0. The difference between our two expressions is then −2y T 1 · H(y 1 ) · y 2 , which depends bilinearily on y 1 and therefore vanishes.
We shall see below (Proposition 3) that the "coordinatewise" affine combination considered in Proposition 1 does have an invariant geometric meaning, provided the coordinate system is geodesic: We say that the metric g on R n (or equivalently, the coordinate system around x 0 ∈ M) is geodesic at 0 ∈ R n (or at x 0 ∈ M, respectively), if the first partial derivatives of G(x), as functions of x ∈ R n , vanish at 0; equivalently, if G(x) = G(0) for every x ∼ 1 0. (This is in turn equivalent to the vanishing at x 0 of the Christoffel symbols of the metric, in the given coordinate system.) It is classical that for every point x 0 , there exists a coordinate system which is geodesic at x 0 . If G(0) is the identity matrix, one talks about a normal coordinate system at x, and such also exist. Cf. e.g. [1] for such notions.
Recall from [5] formula (2) that any Riemannian metric g : M (2) → R admits a unique symmetric extension g : M (3) → R; in coordinates it is given by Recall also from [5] Theorem 3.6 that for x ∼ 2 z in a Riemannian manifold, and for t ∈ R, there exists a unique y 0 with y 0 ∼ 2 x and y 0 ∼ 2 z which is a critical point for the function of y given by tg(x, y) + (1 − t)g(z, y); ( We call this y 0 an (intrinsic) affine combination of x and z. We write it tx + (1 − t)z; this raises a compatibility problem in case we are working in coordinates, since we can then also form the "algebraic" affine combination of two coordinate n-tuples. However, in geodesic coordinates at x, there is no problem, according to the following Proposition, which extends Proposition 3.7 in [5]. Let us consider a coordinate system with x identified with 0.

Proposition 2
The critical point y 0 for the function in (2) is the algebraic affine combination tx + (1 − t)z, if either x ∼ 1 z, or if the coordinate system is geodesic at x.
Proof. Since x is identified with 0 in the coordinate system, the affine combination in question is just (1 − t)z. To show that it is a critical value for (2) means that is independent of v ∼ 1 0. We write g in terms of the symmetric matrices G, as above. Let us take a Taylor expansion of the function G(y), writing where the entries of the matrix H(y) are of degree ≥ 1 in y; and if the coordinate system is geodesic at x = 0, H(y) is even of degree ≥ 2 in y. We then calculate. We get a "significant" part from each of the two terms in (3), and then some "error" terms, each of which will vanish for degree reasons, as we shall argue. The two significant terms are the two terms in Expanding out by bilinearity and symmetry, the terms involving v linearly cancel each other; and the terms involving v quadratically vanish because v ∼ 1 0. So the significant terms, jointly, do not depend on v ∼ 1 0.
The "error" terms are of two kinds: partly, arising from the replacement of G(z) by G(0); here, H(z) enters; and partly there are correction terms when passing from g to g defined on pairs of third order neighbours. The error term of the first kind is a multiple of we expand this out by bilinearity, and use that H(z) is of degree ≥ 1, and v ∼ 1 0. We get four terms each of which vanish for degree reasons if either Finally, the correction terms for upgrading g to g don't occur if z ∼ 1 0, since then g is only applied to pairs of second order neighbours. Thus the assertion of the Proposition about the case z ∼ 1 x is already proved. In general, the upgrading involves first partial derivatives of G, (see (1)), so in the case the coordinate system is geodesic at 0, no correction term is needed for g(0, (1 − t)z), but only for g(z, (1 − t)z + v). Using the formula (1), we see that the correction needed is a certain multiple of hence a linear combination of terms and something that contains v in a bilinear way. All these terms vanish for degree reasons: for, since H vanishes in the first neighbourhood of 0, D v G(z) is of degree ≥ 1 in z, and D z G(z) is even of degree ≥ 2 in z.
Essentially the same degree counting as in this proof gives the following result: Lemma 1 Let y ∼ 1 x and z ∼ 2 x; then using a geodesic coordinate system at x = 0, the quantity g(y, z) may be calculated as (z − y) T · G(0) · (z − y).
Given a Riemannian manifold. If x ∼ 2 z, the mirror image z ′ of z in x is by definition the affine combination 2x − y, i.e. the y which is critical value for 2g(x, y) − g(z, y), [5] Theorem 3.6. Also, the parallelogram formation λ is descibed in [5]. Finally, if t is a tangent vector D → M, its geodesic prolongation t : (Recall that D ⊆ R are the elements of square zero, D 2 the elements of cube zero.) Now Proposition 2 has the following Corollary: Proposition 3 Let x ∼ 2 z; then the mirror image z ′ of y w.r.to x may be calculated as follows: take a geodesic coordinate system at x with x = 0.
Let y ∼ 1 x, z ∼ 1 x. Then λ(x, y, z) may be calculated as follows: take a geodesic coordinate system at x with x = 0. Then λ(x, y, z) = y + z.
Let t be a tangent vector D → M at x ∈ M. Then the geodesic prolongation t : D 2 → M of t may be calculated as follows: take a geodesic coordinate system at x with x = 0. Let u be the unique vector in R n so that t(d) = d · u for all d ∈ D. Then for δ ∈ D 2 , t(δ) = δ · u (The vector u ∈ R n appearing in the last clause is usually called the principal part of t, relative to the coordinate system.) If t and s are tangent vectors at the same point x of a Riemannian manifold M, g, we define their inner product < t, s > by the validity, for all In this way, the tangent vector space T x M is made into an inner product space, (and this is the contact point with the classical formulation of Riemannian metric). If u and v are the principal parts of tangent vectors t and s at x ∈ M, in some coordinate system at x = 0 (not necessarily geodesic), one has < t, s >= u T · G(0) · v; this follows easily from Proposition 1.
Combining Lemma 1 and Proposition 3, one gets Lemma 2 Let t be a tangent vector. Then for d ∈ D, δ ∈ D 2 , we have We are going to define the orthogonal projection of z (z ∼ 2 x) onto a proper tangent t at x. We first define the scalar component of z along t; this is unique number α(z, t) so that for all d ∈ D. Note that if z = x, α(z, t) = 0, and from this follows that for any z ∼ 2 x, α(z, t) ∼ 2 0, in other words α(z, t) ∈ D 2 . From Lemma 2, applied twice (once with d = 0, once with a general d ∈ D), it is immediate to deduce that if z is of the form t(δ) for a δ ∈ D 2 , then α(z, t) = δ. We define the orthogonal projection proj t (z) by Note that it is a second-order neighbour of x. It follows from the above that if z is of the form t(δ), then proj t (z) = z.

Laplacian neighbours
Here is the crucial definition: where n is the dimension of the manifold.
Maybe one of the names "isotropic, harmonic, or conformal, neighbour" would be more appropriate.
Clearly z ∼ 1 x implies z ∼ L x; for if z is a first-order neighbour of x, then so is its orthogonal projection, and hence both the g-quantities to be compared in (6) are zero. If the dimension n is 1, ∼ L is the same as ∼ 2 but in general, the set M L (x) of L-neighbours of x is much smaller than the set M 2 (x) of second-order neighbours; in fact, the ring of functions on M L (x) is a finite dimensional vector space which is just one dimension bigger than the ring of functions on M 1 (x), as we shall see in the proof of Proposition 5 below.
We conjecture that the relation ∼ L is symmetric, but we haven't been able to do the necessary calculations, except in the case of R n , where the symmetry is easy to prove, using Proposition 4 below.
Note the following curious phenomenon in dimension n ≥ 2: if z ∼ L x, then z does not connect to x by any geodesic D 2 → M (given by a proper tangent vector t), except perhaps in the trivial case when g(x, z) = 0. In other words, the L-neighbours of x are genuinely isotropic, in the sense that they are in no preferred direction t (hence the alternative name "isotropic neighbour" suggested). Nevertheless, there are sufficiently many L-neighbours of x to define the Laplacian differential operator ∆, see Theorem 1 below.
Let us assume the manifold in question has dimension n. Then we have Proposition 4 In any geodesic normal coordinate system at Proof. First, if t and s are tangent vectors at x = 0 with principal parts u and v, respectively (meaning for, calculating the enumerator in (5) gives (using Proposition 1) From the third clause in Proposition 3, we then get the familiar looking In particular, if t is the (proper) tangent vector with principal part e i ∈ R n (= (0, . . . , 1, . . . 0) (with 1 in the i'th position, 0's elsewhere), then proj t (z 1 , . . . , z n ) = z i e i . In particular g(0, proj t (z)) = z 2 i . If, on the other hand, t is the tangent vector with principal part e i,j (the vector with 1's in the i'th and in the j'th position, i = j, 0's elsewhere), then proj t (z) has (z i +z j )/2 in the i'th and in the j'th position, and 0's elsewhere. In particular, If z therefore is an L-neighbour of 0, we conclude that z 2 i = z 2 j for all i, j, and that z i z j = 0 if i = j.
Conversely, assume that in some geodesic normal coordinate system at x = 0, the coordinates (z 1 , . . . , z n ) satisfy the equations z 2 i = z 2 j , z i z j = 0 for i = j, and let t be a proper tangent vector at x with principal part u = (u 1 , . . . , u n ). Then which we calculate by arithmetic to be but since z i z j = 0 for i = j, only the "diagonal" terms survive, and we are left with But z i z i = z 1 z 1 for all i, so this factor can go outside the sum sign in the enumerator, and we get z 2 1 ( i u i u i )/u • u = z 2 1 , which is 1/n times z 2 i since all the z 2 i are equal. This proves the Proposition. From Propositions 3 and 4, one immediately deduces that if z ∼ L x, then also z ′ ∼ L x for any affine combination z ′ = tx + (1 − t)z (t ∈ R).

Proposition 5
If two functions f 1 and f 2 : Proof. Using a geodesic normal coordinate system at x = 0, it is a matter of analyzing the ring of functions M L (0) → R for the case where M = R n with standard inner-product metric. The Proposition gives that M L (0) may be described as D L (n) ⊆ R n , defined by (for n ≥ 2; for n = 1, D L (1) = D 2 = {δ ∈ R | δ 3 = 0}). This is (for n ≥ 2) the object represented by the Weil algebra O(D L (n)) := k[Z 1 , . . . , Z n ]/I, where I is the ideal generated by the Z 2 i − Z 2 j , and by Z i Z j for i = j. It is immediate to calculate that, as a vector space, this ring is (n+2)-dimensional, with linear generators 1, Z 1 , . . . , Z n , Z 2 1 + . . . + Z 2 n .
By the general (Kock-Lawvere) axiom scheme for SDG [4], [8], this means that any function f : for unique a, b 1 , . . . , b n , c ∈ R, or equivalently Since the restriction of f to D(n) is given by the data a, b 1 , . . . , b n , the unique existence of c follows.
The following Theorem deals with an arbitrary Riemannian manifold M, g of dimension n, and gives a coordinate free characterization of the Laplacian operator ∆.
Theorem 1 For any f : M L (x) → R, there is a unique number L with the property that for any z ∼ L x where z ′ denotes the mirror image of z in x. We write ∆f (x) := nL.
Put differently, If the function f is harmonic at x, meaning that ∆f (x) = 0, it follows that it has a strong average value property: the value at x equals the average value of f over any pair of points z and z ′ (L-neighbours of x) which are symmetrically located around x.
Proof. Again, we pick a normal geodesic coordinate system at x = 0, so identify M L (x) with D L (n); then z ′ gets identified with −z, by Proposition 3. The left hand side of the expression in the Theorem then has restriction 0 to D(n), being (with notation as above) (a + b i z i ) + (a + b i (−z i )) − 2a.
Hence the unique existence of L follows from Proposition 5.
The following Proposition serves to as partial justification of the use of the name "Laplacian" for the ∆ considered in the Theorem. We consider the standard Riemannian metric on R n , g(x, z) = ||z − x|| 2 , for z ∼ 2 x.
Proposition 6 Let f : R n → R and let x ∈ R n . Then Proof. For simplicity, let x = 0, so that z ′ = −z. We Taylor expand f (z) and f (−z) from 0, and consider f (z) + f (−z) − 2f (0); then terms of degree ≤ 1 cancel, and we get ij ∂ 2 f ∂x i ∂x j z i z j + higher terms; now the calculation proceeds much like the one in the proof of Proposition 4 above: if z ∼ L 0, only the diagonal terms in the sum survive, all the z 2 i are equal to z 2 1 , which we move outside the parenthesis, and get z 2 1 times the classical Laplacian ∂ 2 f /∂x 2 i (0). But z 2 1 = 1/n · g(0, z). Similarly, one proves by Taylor expansion Proposition 7 If z ∼ L x in R n , then for any f : R n → R, Recall that any function M → R looks affine on any 1-monad M 1 (x); functions that look affine on the larger L-monads M L (x) are precisely the harmonic ones: Proof. Assume z ∼ L x, and pick a geodesic normal coordinate system at x = 0. Without loss of generality, we may assume f (0) = 0. Then to say that f preserves affine combinations of x and z is to say that for all s ∈ R, f (sz) = sf (z). For z ∼ L 0, we have by Proposition 7 that f (z) = a i z i + c z 2 i for unique a i and c (c = ∆f (0)/2n); f (sz) and sf (z) have the same terms of first order in z; their second order terms are respectively cs 2 z 2 i and sc z 2 i , and if these two expressions are to be equal for all s and all z, we must have c = 0. This means that f is harmonic at x. Conversely, if f preserves affine combinations of the kind mentioned, it preserves the affine combination 2x − z, or equivalently, the left hand side of the expression in Theorem 1 is 0, hence it follows that L and hence ∆f (x) is 0.
Remark. With some hesitation, I propose to call a map f : M → N between Riemannian manifolds harmonic if it preserves affine combinations of L-neighbours in M and if it preserves the property of being L-neighbours. I have not been able to compare the proposed definition, with a certain classical concept of harmonic map between Riemannian manifolds. But at least: When the codomain is R, the definition is the basic classical one of harmonic function, by Proposition 8. For, preservation of the ∼ L relation is automatic when the codomain is R, since in R, ∼ L is the same as ∼ 2 .

Conformal maps
We consider a diffeomorphism f : M → N between Riemannian manifolds (M, g), (N, h). To say that f is an isometry at x ∈ M is to say that for f is an isometry at x). The terminology agrees with classical usage, as we shall see below. We first prove z). Then for all y 1 ∼ 1 x, y 2 ∼ 1 x, h(f (y 1 ), f (y 2 )) = kg(y 1 , y 2 ), and conversely.
Proof. We choose coordinates, and assume x = 0 and f (x) = 0; the metrics in M and N are then given by functions g and h, respectively, and they are in turn given by symmetric matrices G(y), and H(z) for all y ∈ M and z ∈ N. We now calculate kg(y 1 , y 2 ). We have, by Proposition 1 that kg(y 1 , y 2 ) = kg(0, y 2 − y 1 ) = h(0, f (y 2 − y 1 )). Now there is a bilinear B(−, −) such that for all pairs of 1-neighbours y 1 , y 2 of 0, we have f (y 2 − y 1 ) = f (y 2 ) − f (y 1 ) + B(y 1 , y 2 ). So the calculation y 2 )).
Since f depends in a linear way of y 1 ∼ 1 0 and y 2 ∼ 1 0, this whole expression multiplies out by linearity, and for degree reasons all terms involving B, as well as some others, vanish, and we are left with −2f (y 1 ) T · H(0) · f (y 2 ). On the other hand, h(f (y 1 ), h(f (y 2 )) = h(0, f (y 2 ) − f (y 1 )), by Proposition 1, and writing this in terms of H(0) gives the same expression. The converse is proved in the same way for z ∼ 2 0 of the form y 2 − y 1 with y 1 ∼ 1 0 and y 2 ∼ 1 0, but this suffices to get the result for all z ∼ 2 0, by general principles of SDG ("R, and hence any manifold, perceives the addition map D(n) × D(n) → D 2 (n) to be epic".) Call a linear map F : U → V between inner product vector spaces conformal with constant k > 0 if for all u 1 , u 2 ∈ U < F (u 1 ), F (u 2 ) >= k < u 1 , u 2 > .
It follows immediately from Proposition 9, and from the construction of inner product in the vector space of tangents at x, and at f (x), that if f is conformal at x with constant k, then df x : T x M → T f (x) N is a conformal linear map with the same constant k. The converse also holds; for if df x is conformal with constant k, we deduce that for all pairs of tangents t and s at x g(f (t(d 1 )), f (s(d 2 ))) = k · g(t(d 1 ), s(d 2 )), and hence g(f (y 1 )), f (y 2 )) = k · g(y 1 , y 2 ) for all y i 's of the form t(d) for a tangent vector t and a d ∈ D. Again by general principles, any manifold N "perceives all 1-neighbours of x to be of this form". From Proposition 9 we therefore deduce that f is conformal at x with constant k.
Proof. Assume f maps M L (x) into M L (f (x)). We may pick normal coordinates at x as well as at f (x). The neighbourhoods M L (x) and M L (f (x)) then both get identified with D L (n), and x = 0, f (x) = 0. The restriction of f to D 2 (n), f : D 2 (n) → R n , takes 0 to 0 and is therefore of the form f (y) = A · y + B(y), where A is an n × n matrix, and B(y) is a map R n → R n which is homogeneous of degree 2 in y ∈ R n , i.e. an n-tuple of quadratic forms B i .
Assume now that f maps D L (n) into itself. For z ∈ D L (n), the i'th Squaring this, only the terms in ( k a ik z k )( l a il z l ) survive for degree reasons (using that z ∈ D 2 (n)). But using further that z k z l = 0 for k = l, only the "diagonal" terms survive, and we get Similarly If now f (z) ∈ D L (n) for all z ∈ D L (n), we get that the expression in (10) is independent of i, and from the uniqueness assertion in Proposition 5 we therefore conclude k a 2 ik = k a 2 jk for all i, j; and similarly we conclude from (11) that k a ik a jk = 0 for i = j.
These two equations express that all the rows of the matrix A have the same square norm k, and that they are mutually orthogonal. This implies that the linear map df x represented by the matrix is conformal, and hence f is conformal at x. The proof that conformality of f at x implies that f maps M L (x) into M L (f (x)) goes essentially through the same calculation, and is omitted.

A famous pseudogroup in dimension 2
The content of the present section is partly classical, namely the equivalence of the various ways of describing the notion of holomorphic map from (a region in) the complex plane C = R 2 to itself. Synthetic concepts enter essentially in two of the conditions in the Theorem below, namely 1) and 7).
An almost complex structure on a general manifold M consists in giving, for each x ∈ M, a map I x : M 1 (x) → M 1 (x) with I x (x) = x and I x (I x (z)) = z ′ for any z ∼ 1 x; Here, z ′ denotes the mirror image of z in x, i.e. the affine combination 2x − z; recall [7] that affine combinations of 1-neighbours make "absolutely" sense, i.e. do not depend on, say, a Riemannian structure. It is clear what it means for a map f to preserve such structure at the point x: The manifold R 2 carries a canonical almost-complex structure, given by Identifying R 2 with the complex plane C, this is just Utilizing the multiplication of the complex plane C, we may consider the set D C of elements of square zero in C (recalling the fundamental role which the set D of elements of square zero in R plays in SDG). We have, by trivial calculation, Proposition 10 Under the identification of C with R 2 , Having D C , we may mimick the basics of SDG and declare a function f : (The uniqueness of such f ′ (x), justifying the notation, follows from the general axiom scheme of SDG, applied to D 1 (2), the 1-jet classifier in R 2 . Note D 1 (2) ⊆ D L (2).) -The notion of course makes sense for functions f which are just defined locally around x.
Theorem 3 Let f : R 2 → R 2 be a (local) orientation preserving diffeomorphism. Let x ∈ R 2 = C. Then the following conditions are equivalent: f satisfies Cauchy-Riemann equations at x 4) f preserves almost complex structure at x. Also the following conditions are equivalent, and they imply 1)-4): ), and f preserves affine combinations of x and z for any z ∼ L x. Finally, if 1)-4) hold for all x, 5) and 6) hold for all x.
(Note that 6) says that f is harmonic at x, in the sense of Remark at the end of Section 2.) Proof. The equivalence of 1) and 2) is already in Theorem 2, and this in turn is, as we have seen, equivalent to conformality of the linear df x . But conformal orientation preserving 2 × 2 matrices are of the form Since the entries of the matrix for df x are ∂f i /∂x j , this form (12) of the matrix therefore expresses that the Cauchy-Riemann equations hold at x, i.e. is equivalent to 3). On the other hand, a simple calculation with 2 × 2 matrices give that a matrix commutes with the matrix I = 0 −1 1 0 for the almost complex structure iff it has the above "Cauchy-Riemann" form (12). Now assume 5). If f is complex differentiable at x, we prove that condition 1) holds at x as follows. Let z ∼ L x. Then (z − x) 2 = 0 Proposition 10, and by complex differentiability so since the right hand side has square zero, then so does the left hand side, but again by Proposition 10, this means that f (z) ∼ L f (x), proving 1), and hence also the first part of 6). But also, if f is complex-differentiable at x, f preserves affine combinations of the form tx + (1 − t)z for z ∈ M L (x); this follows from (13), since z − x ∈ D C (x) by Proposition 10, so also the second part of 6) is proved. Conversely, if 6) holds, f is conformal at x by Theorem 2, so df x is if the form (12). Then f ′ (x) = a + ib will serve as the complex derivative; for, since f preserves affine combinations of x and z for z ∼ L x, we have the first equality sign in for such z, i.e. for z − x ∈ D L (2) = D C . Finally, assume 1)-4) hold for all x. Then we may differentiate the Cauchy-Riemann equations for f = (f 1 , f 2 ) by ∂/∂x 1 and ∂/∂x 2 and compare, arriving in the standard way to ∆f 1 ≡ 0 and ∆f 2 ≡ 0. From the "Taylor expansion" in Proposition 7, applied to f 1 , we conclude that f 1 (z) = f 1 (x)+(df 1 ) x (z−x) for z ∼ L x, and similarly for f 2 , so f (z) = f (x)+df x (z−x) for such z, i.e. for z − x ∈ D C . Since df x is given by a conformal matrix (12), by 2), this proves that a + ib will serve as the complex derivative of f at x.

Support of the Laplacian
We arrived at D L (n) from the geometric side, namely as the ∼ L -neighbours of 0 in the Riemannian manifold M = R n ; the differential operator ∆ was then seen to provide the top term in the Taylor expansion of functions defined on D L (n).
Here, we briefly indicate how to arrive at D L (n) from the algebraic side, starting with ∆ = ∂ 2 /∂x 2 i . More precisely, we consider ∆ as a distribution at 0 ∈ R n . So ∆ is the linear map k[X 1 , ..., X n ] → R given by The algebraic concept that will give D L (n) out of this data is the notion of coalgebra, and subcoalgebra, as in [9]. If we let A denote the algebra k[X 1 , ..., X n ], then the distribution ∆ of (14) factors where A → B is an algebra map, and B is finite dimensional (take e.g. B = A/J where J is the ideal generated by monomials of degree ≥ 3). The set A o of linear maps A → k having such a factorization property constitute a coalgebra, [9] Proposition 6.0.2. Every element in a coalgebra generates a finite dimensional subcoalgebra, by [9] Theorem 2.2.1. In particular, ∆ ∈ A o generates a finite dimensional coalgebra [∆] of A o , and this coalgebra "is" D L (n). More specifically, the dual algebra of [∆] is the coordinate ring O(D L (n)) of D L (n), i.e. the Weil algebra O(D L (n)) := k[Z 1 , ..., Z n ]/I considered in the proof of Proposition 5, as we shall now argue. The following "Leibniz rule" for ∆ is well known, This means that in the coalgebra A o , we have the following formula for ψ(∆) (ψ = the comultiplication of the coalgebra; δ the Dirac distribution "evaluate at 0"): where now ∆, ∂d/∂x i , δ are viewed as distributions at 0, like in (14), meaning that one evaluates in 0 after application, From (15), (and from ψ(∂/∂x i ) = ∂/∂x i ⊗ δ + δ ⊗ ∂/∂x i , which expresses the Leibniz rule for ∂/∂x i ) we see that the subcoalgebra [∆] generated by ∆ is generated as a vector space by the elements δ, ∂ ∂x 1 , ..., ∂ ∂x n , ∆, and since these are clearly linearly independent, we see that [∆] ⊆ A o is (n + 2)-dimensional. The dual algebra of [∆] is a quotient algebra A/I of A, where I is the ideal of those f ∈ A which are annihilated by the elements of [∆]. This ideal I contains x 2 i − x 2 j , and x i x j for i = j. Since the quotient of A by the ideal generated by x 2 i − x 2 j , and x i x j for i = j is already (n + 2)dimensional, as calculated in the proof of Proposition 5, it follows that the quotient algebra there is actually the dual of [∆].
The idea that a coalgebra like [∆] is itself an infinitesimal geometric object goes back to Gavin Wraith in the early seventies, [10]. The specific way of generating Weil algebras from differential operators was considered by Emsalem [2] (without coalgebras).