A Nonstandard Analysis Characterization of Submanifolds in Euclidean Space

Udo Hertrich-Jeromin

Abstract

A nonstandard analysis characterization for standard submanifolds in a Euclidean space is given.

Mathematics Subject Classification: 53A07 Key words: nonstandard analysis, submanifolds, Nelson approach

1 Introduction

The aim of this note is to prove a nonstandard analysis characterization for C^{1}- submanifolds in Euclidean n-space. This characterization features a purely geometric approach that might be useful in the context of “infinitesimal geometry”, cf. [1]. In Stroyan’s fundamental paper [6], a weaker statement than the one we are going to formulate is proven — even though some remarks strongly suggested the characterization given here^{1}. In the author’s opinion, it should be possible to give similar characterizations for higher order differentiable submanifolds. However, the elementary methods used here seem not to be appropriate to produce a rigorous proof.

We are going to formulate our statement (and its proof) using Nelson’s approach [2] to nonstandard analysis (cf.[4]): thus we work with the usual objects of “classical” (standard) mathematics, and all statements of classical mathematics are valid in our setting. We just deal with an additional attribute “standard” for mathematical objects that allows to define terms like “infinitely close”, or “infinitesimal”. However, this attribute has to be used with care as it behaves different than the attributes one is used to from classical mathematics. For more details, the reader is referred to [4], [1], or the original paper [2] by E. Nelson^{2}.

Before formulating our theorem, we give a useful Notation. We say that “q ∈ R^{n} is infinitely close to p ∈ R^{n} mod μ”, q ≃_{μ} p, if

Balkan Journal of Geometry and Its Applications, Vol.6, No.1, 2001, pp. 15-22 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

1This misled the author in [1] to erroneously cite Stroyan’s paper for the theorem proven in the present paper. And, after noticing this error, the author was very surprised that he could not find any alternative reference for our theorem below.

2Also, there exists an excellent part of an unwritten book by E. Nelson [3].

\frac{|q − p|}{|μ|} ≤ ε for all standard ε > 0 (or, q = p in case μ = 0).

In case μ = 1, we write q ≃ p, and say that “q is infinitely close to p”.

With this, we formulate the main Theorem. A standard subset M^{m} ⊂ R^{n}, n ≪ ∞ a positive standard integer, is a C^{1}- submanifold if and only if there exists a standard tangent plane map T : M → G(m, n) into the set of affine m-planes such that, for every near standard (in M) point p ∈ M,

(a) p lies on its tangent plane^{3}, p ∈ T_{p};

(b) the orthogonal projection π_{p} : M → T_{p} is an infinitesimal bijection^{4};

(c) the angle that the secant line through p and any infinitely close point q ∈ M, q = p + dp ≃ p, forms with T_{p} is infintesimal: |q − π_{p}(q)| ≃_{|dp|} 0.

2 Some linear algebra

A key issue in the proof of the theorem is to control the “angle” between nearby tangent planes. As we are dealing with arbitrary dimension and codimension, it seems necessary to first clarify this notion of angle: for that purpose, note that the Euclidean scalar product on R^{n} uniquely extends to the Grassmann algebra ΛR^{n} = ⊕_{k=0}^{n}Λ^{k}R^{n} of R^{n} via ⟨v_{1} ∧…∧v_{k}, w_{1} ∧…∧w_{k}⟩ = det(⟨v_{i}, w_{j}⟩)_{ij}, and Λ^{m}R^{n} ⊥ Λ^{k}R^{n} for m ≠ k. Lemma & Definition. Let V, W ⊂ R^{n} be two m-dimensional subspaces, and choose bases (v_{1}, …, v_{m}) and (w_{1}, …, w_{m}). Then, the number α ∈ [0, \frac{π}{2} ] with

|v_{1} ∧ … ∧ v_{m}| |w_{1} ∧ … ∧ w_{m}| cos α = |⟨v_{1} ∧ … ∧ v_{m}, w_{1} ∧ … ∧ w_{m}⟩|

does not depend on the choice of bases for V and W. We will call α the “intersection angle of V and W ”. Proof. First note that α is well defined, by the Cauchy-Schwarz inequality. To understand that α is independent of the choice of bases, consider an endomorphism

A ∈ End(V), ṽ_{i} = ∑_{j=1}^{m} a_{ij}v_{j}. Then,

\begin{aligned}⟨ṽ_{1} ∧ … ∧ ṽ_{m}, w_{1} ∧ … ∧ w_{m}⟩ & = & det(∑_{j=1}^{m} a_{ij}⟨v_{j}, w_{k}⟩)_{ik} \\ & = & det(A) · ⟨v_{1} ∧ … ∧ v_{m}, w_{1} ∧ … ∧ w_{m}⟩ \end{aligned}

which implies the claim.

Using that description of the just defined intersection angle of subspaces as an ordinary angle of two vectors in a higher dimensional vector space, we obtain a triangle inequality: Lemma 1. Given three m-dimensional subspaces V_{i} ⊂ R^{n}, i = 1, 2, 3, with intersection angles α_{ij}, we have α_{13} ≤ α_{12} + α_{23}. Proof. Thinking of the subspaces V_{i} as lines in Λ^{m}R^{n} ≅ R^{( {n \atop m})}, this just is the triangle inequality in RP^{( {n \atop m})} equipped with the spherical metric. □

3As this is a classical property, it will hold for all p ∈ M as soon as it holds for all standard p, by transfer.

4This means: for M ∋ q, q̃ ≃ p, π_{p}(q) = π_{p}(q̃) implies q = q̃ (i.e., π_{p} is an infinitesimal injection), and for any T_{p} ∋ x ≃ p there is a q ∈ M, q ≃ p, with π_{p}(q) = x (π_{p} is an infinitesimal surjection).

A Nonstandard Analysis Characterization of Submanifolds in Euclidean Space 17

On the other hand, it will be important to relate the intersection angle α between two subspaces with angles of vectors spanning them. Lemma 2. If V, W ⊂ R^{n} are two m-dimensional subspaces, then there exist orthonormal bases (v_{1}, …, v_{m}) and (w_{1}, …, w_{m}) of V and W, respectively, such that

⟨v_{i}, w_{j}⟩ = δ_{ij} cos α_{i} with angles α_{i} ∈ [0, \frac{π}{2} ].

Proof. First, choose an orthonormal basis (v_{1}, …, v_{k}) of V ∩ W, and let w_{i} := v_{i} for i = 1, …, k. Then, denote S := {x ∈ R^{n} | |x| = 1 & x ⊥ V ∩ W }, and consider the function^{5} f : (V ∩ S) × (W ∩ S) =: K → R, (v, w) ↦ f(v, w) = ⟨v, w⟩. As K is compact, there are unit vectors v_{k+1} ∈ V and w_{k+1} ∈ W such that f(v_{k+1}, w_{k+1}) < 1 is maximal. Now, let w ∈ W, w ⊥ w_{k+1}; then, w ⊥ v_{k+1}: since s ↦ f(v_{k+1}, w_{k+1} cos s + w sin s) becomes maximal for s = 0, its derivative at s = 0 vanishes so that ⟨v_{k+1}, w⟩ = 0. By symmetry, the same is true for v ∈ V, v ⊥ v_{k+1}. Thus, taking S = {x ∈ R^{n} | |x| = 1 & x ⊥ V ∩ W, v_{k+1}, w_{k+1}}, we may reduce the dimension by 2 × 1. Repeating this construction, we get bases of V and W as desired.

□

In particular^{6}, cos α = ∏_{k=1}^{m} cos α_{k}. As an immediate consequence, we obtain the

following Lemma 3. The orthogonal projection π : R^{n} → V satisfies |π(w)| ≥ |w| cos α for all w ∈ W, where α denotes the intersection angle of V and W.

And^{7}, the restriction π|_{W} : W → V of π yields an isomorphism as soon as cos α > 0, or α ≠ \frac{π}{2} : Lemma 4. Let V, W ⊂ R^{n} be two m-dimensional linear subspaces with intersection angle α ∈ [0, \frac{π}{2} ], and denote π : R^{n} → V the orthogonal projection of R^{n} onto V;

then, the restriction π|_{W} : W → V is an isomorphism iff α < \frac{π}{2} .

Finally, we will need two rather technical estimates that will take care of some substantial part of the proof of our theorem. We prove them here. Lemma 5. Let V, W ⊂ R^{n} be two m-dimensional linear subspaces with intersection angle α < \frac{π}{4} , and denote π : R^{n} → V the orthogonal projection of R^{n} onto V; let

x ∈ R^{n} make the angle β < \frac{π}{4} with its orthogonal projection onto W, and denote ^{8}

v := π(x) and w := π|_{W}^{−1}(π(x)). Then,

(i) |x − w| ≤ \frac{1}{cos^{2} α(1 − tan α · tan β)} tan β · |v|, and

(ii) |x − v| ≤ tan(α + β) · |v|.

5Note that f(v, w) = cos^{}(v, w); thus, maximizing f is equivalent to minimizing the (spherical)

distance. Also, note that f(v, −w) = −f(v, w): thus, since w ∈ S ∩ W ⇔ −w ∈ S ∩ W, the maximum of f is ≥ 0, and the spherical distance is ≤ \frac{π}{2} .

6Thus, the intersection angle of two subspaces is not the Hausdorff distance of the two (m − 1)- spheres S^{n−1} ∩ V and S^{n−1} ∩ W, i.e., it is, generally, not the “worst angle” between V and W.

However, in the cases of dimension m = 1, or codimension n − m = 1, the angle α is exactly what one expects to be the intersection angle of two lines resp. hyperplanes.

7Another proof is by noticing that, given any orthonormal bases (v_{1}, …, v_{m}) and (w_{1}, …, w_{m}) of V and W, respectively, the matrix (⟨w_{i}, v_{j}⟩)_{ij} is exactly the representation matrix of π|_{W} : W → V.

Hence, rk(π|_{W}) = m if and only if α < \frac{π}{2} .

8Note that π|_{W} : W → V is an isomorphism as α < \frac{π}{2} .

18 Udo Hertrich-Jeromin Proof. Denote Y := (V ⊕ W)^{⊥} and Z := (V ⊕ W) ∩ (V ∩ W)^{⊥}; note that Z has even dimension and dim(V ∩ Z) = dim(W ∩ Z) = \frac{1}{2}dimZ =: k. Thus, as above, we can choose bases (v_{1}, …, v_{k}) and (w_{1}, …, w_{k}) of V ∩ Z and W ∩ Z such that span{v_{1}, …, v_{k}, w_{1}, …, w_{k}} = Z and ⟨v_{i}, w_{j}⟩ = cos α_{j} · δ_{ij}. Then, we have cos α = ∏_{j=1}^{k} cos α_{j}. Further, let v_{j}^{⊥} and w_{j}^{⊥}, j = 1, …, k, denote vectors such that w_{j} = cos α_{j} · v_{j} + sin α_{j} · v_{j}^{⊥} and w_{j}^{⊥} = − sin α_{j} · v_{j} + cos α_{j} · v_{j}^{⊥}. With this setup, we write

x = x_{0} + ∑_{j=1}^{k}⟨x, v_{j}⟩v_{j} + ∑_{j=1}^{k}⟨x, v_{j}^{⊥}⟩v_{j}^{⊥}  + y
v = x_{0} + ∑_{j=1}^{k}⟨x, v_{j}⟩v_{j}
w = x_{0} + ∑_{j=1}^{k}⟨x, v_{j}⟩v_{j} + ∑_{j=1}^{k} tan α_{j}⟨x, v_{j}⟩v_{j}^{⊥}

since π|_{W}^{−1}(v_{j}) = v_{j} + tan α_{j} · v_{j}^{⊥}. With tan^{2} β = \frac{|y|^{2} + ∑⟨x, w_{j}^{⊥}⟩^{2}}{|x_{0}|^{2} + ∑⟨x, w_{j}⟩^{2}} we compute

|x − w|^{2} = |y|^{2} + ∑_{j=1}^{k}⟨x, − tan α_{j} · v_{j} + v_{j}^{⊥}⟩^{2}
= tan^{2} β · |π̃(x)|^{2} + ∑_{j=1}^{k} tan^{2} α_{j}⟨x, w_{j}^{⊥}⟩^{2}
≤ (1 + tan^{2} α) tan^{2} β · |π̃(x)|^{2}

where π̃(x) = x_{0} + ∑_{j=1}^{k}⟨x, w_{j}⟩w_{j} is the orthogonal projection of x onto W. Since w_{j} =

\frac{1}{cos α_{j}} v_{j} +tan α_{j} ·w_{j}^{⊥}, we deduce (|π̃(x)|−|w|)^{2} ≤ |π̃(x)−w|^{2} = ∑_{j=1}^{k} tan^{2} α_{j}⟨x, w_{j}^{⊥}⟩^{2} ≤

tan^{2} α · tan^{2} β · |π̃(x)|^{2} which gives

|π̃(x)| ≤ \frac{1}{1 − tan α · tan β} |w| ≤ \frac{1}{(1 − tan α · tan β) cos α}|v|

to obtain the first estimate. The second is obtained from

|w − v|^{2} = ∑_{j=1}^{k} tan^{2} α_{j}⟨x, v_{j}⟩^{2}
≤ tan^{2} α ∑_{j=1}^{k}⟨x, v_{j}⟩^{2}
≤ tan^{2} α · |v|^{2}

via the triangle inequality |x − v| ≤ |x − w| + |w − v|. □

However, we consider the affine tangent plane map of a submanifold. Thus, we have to clarify what the angle between two affine m-planes in R^{n} should be — here, one major issue is that two affine m-planes need not intersect. Definition. Let p + V, q + W ⊂ R^{n} be two affine m-planes with direction vector spaces V and W, respectively. We define the “angle α ∈ [0, \frac{π}{2} ] between p + V and q + W ” to be the intersection angle of V and W.

Note that the angle defined this way does not depend on the choice of base points p and q, hence defines an “angle between the two m-planes”, indeed. The properties of the intersection angle of two linear subspaces proven above carry over for the angle between two (affine) m-planes — in some cases, minor and obvious reformulations are required that we leave to the reader.

3 Proof of the Theorem

First, let M ⊂ R^{n}, M and n standard, be a C^{1}-submanifold. By transfer, the (affine) tangent plane map p ↦ T_{p} (and hence, the map p ↦ N_{p} assigning to each point the affine normal space of M) is standard — taking standard values at standard points. Clearly, the tangent plane map T satisfies (a).

Now, let q ∈ M be near standard, and p ∈ M standard with p ≃ q. As a C^{1}- submanifold, M can locally be represented as a graph over its tangent plane^{9} at p, i.e., there exists a locally defined C^{1}-function g : T_{p} ⊃ U → U^{⊥} ⊂ N_{p} such that

M ∩ (U × U^{⊥}) = {(x, g(x)) ∈ (T_{p} × N_{p}) ≅ R^{n} | x ∈ U}.

As the existence of g is a classical statement with standard parameters p, M, T_{p} and N_{p}, the function g (and hence, U and U^{⊥}) can be assumed to be standard, by transfer. In particular, for every q ∈ M, q ≃ p, there is a unique x ∈ T_{p} such that q = (x, g(x)), i.e., π_{p}(q) = x.

Since M is C^{1}, we can, after possibly making U smaller, assume that the angle^{10} α_{p,q} between any tangent plane T_{q}, q ∈ M ∩ (U × U^{⊥}), and T_{p} is less than \frac{π}{4} . This is a classical statement, such that U can still be assumed to be standard. Then, by the triangle inequality (Lemma 1), the angle between any two tangent planes at points in M ∩ (U × U^{⊥}) is less than \frac{π}{2} . Hence, M ∩ (U × U^{⊥}) intersects the fibres of any orthogonal projection π_{q} : R^{n} → T_{q} transversally, and, consequently, π_{q}|_{M∩(U×U^{⊥})} : M ∩ (U × U^{⊥}) → T_{q} is a bijection onto its image. Thus, for q ≃ p, π_{q} yields an infinitesimal bijection. This proves (b).

Finally, to show (c), let q + dq ∈ M be infinitely close to q, q + dq ≃ q, and write q = (x, g(x)) and q+dq = (x+dx, g(x+dx)) in terms of the above local representation of M as a standard graph. Since π_{q}(q + dq) ∈ T_{q} minimizes, in T_{q}, the distance to q + dq we have

9This is a straightforward application of the implicit mapping theorem.

10Note that the map q ↦ α_{p,q} is continuous since g is continuously differentiable.

|(q + dq) − π_{q}(q + dq)| ≤ |(x + dx, g(x + dx)) − ((x, g(x)) + (dx, d_{x}g(dx)))|
= |g(x + dx) − g(x) − d_{x}g(dx)|
= | ∫_{0}^{1} d_{x+t·dx}g(dx) − d_{x}g(dx)dt|
≤  (∫_{0}^{1} |d_{x+t·dx}g − d_{x}g|dt) · |dx| ≃_{|dx|} 0

since g is continuously differentiable, i.e., max_{t∈[0,1]}{|d_{x+t·dx}g − d_{x}g|} ≃ 0. On the other hand, dq = (x + dx, g(x + dx)) − (x, g(x)) ≃_{|dx|} (dx, d_{x}g(dx)) such

that 0 ≪ \frac{|dq|}{|dx|}, that is, 0 < \frac{|dq|}{|dx|} but 0 ≄ \frac{|dq|}{|dx|}; hence, (…) ≃_{|dx|} 0 ⇒ (…) ≃_{|dq|} 0,

and (c) follows.

Conversely, assume that a standard subset M ⊂ R^{n}, n ≪ ∞, satisfies conditions (a) through (c) of the theorem. Let p ∈ M be standard.

First, we want to show that M can be written as the graph of a standard function g over its tangent plane T_{p}. Denoting B_{δ}(p) ⊂ T_{p} and B_{δ}^{⊥}(p) ⊂ N_{p} the open δ-ball around p in the (affine) tangent and normal space of M at p, respectively, (b) tells us that the orthogonal projection π_{p} : R^{n} → T_{p} is a bijection from M ∩ (B_{δ}(p) × B_{δ}^{⊥}(p)) onto its image for any infinitesimal δ > 0. By (c), for q = p+dp ∈ M ∩(B_{δ}(p)×B_{δ}^{⊥}(p)), we have q −π_{p}(q) ≃_{|dp|} 0: hence, |q −π_{p}(q)| < δ, as |dp| < \sqrt{2} δ, and π_{p} : M ∩(B_{δ}(p)× B_{δ}^{⊥}(p)) → B_{δ}(p) is onto. By the overspill principle, there is a standard δ > 0 such that, with U := B_{δ}(p) and U^{⊥} := B_{δ}^{⊥}(p), π_{p} : M ∩(U ×U^{⊥}) → U is a bijection. Thus, we can write M ∩ (U × U^{⊥}) as a graph of the function g = π_{p}|_{M∩(U×U^{⊥})}^{−1} : U → U^{⊥}, i.e.,

M ∩ (U × U^{⊥}) = {(x, g(x)) | x ∈ U}.

Next, we want to show that g is C^{1}. To that end, we first show that the (standard, as p is standard) map q ↦ α_{p,q}, assigning to each point q ∈ M the angle between the tangent planes T_{q} and T_{p}, is continuous. Thus, let q ∈ M be standard and q+dq ∈ M, q + dq ≃ q. Since, by the spherical triangle inequality (Lemma 1), α_{p,q+dq} ≤ α_{p,q} + α_{q,q+dq} as well as α_{p,q} ≤ α_{p,q+dq} + α_{q+dq,q}, it suffices to show that α_{q,q+dq} ≃ 0 whenever q is standard and dq ≃ 0: Lemma 6. The angle between the tangent planes T_{q} and T_{p} at two infinitely close, near standard points p, q ∈ M is infinitesimal. Proof. Denote δ := |q − p|, choose some direction in T_{q}, and let q̃ ∈ M be such that |q̃ − q| = δ and t_{q} := π_{q}(q̃) − q yields the chosen direction (this is possible by the bijection assumption (b)). Further, decompose t_{q} = t_{p} + n_{p} into components t_{p} and n_{p} parallel and orthogonal to T_{p}, respectively. This way we have, by (c), q̃ ≃_{δ}

π_{q}(q̃) as well as q̃ ≃_{δ} π_{p}(q̃) since 0 ≃ \frac{|q̃ − π_{p}(q̃)|}{|q̃ − p|} ≥ \frac{|q̃ − π_{p}(q̃)|}{|q̃ − q| + |q − p|} = \frac{|q̃ − π_{p}(q̃)|}{2δ}.

Then, q + t_{q} = π_{q}(q̃) ≃_{δ} q̃ ≃_{δ} π_{p}(q + t_{q}) as |π_{p}| ≤ 1 and q̃ ≃_{δ} q + t_{q}. Using the decomposition t_{q} = t_{p} + n_{p} and the fact that π_{p} is an affine map, we arrive at π_{p}(q + t_{q}) = π_{p}(q) + t_{p} ≃_{δ} q + t_{p}. Thus, t_{q} ≃_{δ} t_{p}. For the angle formed by t_{q} and t_{p},

we conclude that \frac{⟨t_{p}, t_{q}⟩}{|t_{p}| |t_{q}|} = \frac{|t_{p}|}{|t_{q}|} ≃ 1 as δ = |q̃ − q| ≃_{δ} |t_{q}| ≃_{δ} |t_{p}|.

But, t_{q} was an arbitrary direction in T_{q}; hence, the above basis representation

(Lemma 2) for the angle yields the claim^{11}: cos α = ∏_{j=1}^{m} \frac{⟨(t_{q})_{i}, (t_{p})_{i}⟩}{|(t_{q})_{i}| |(t_{p})_{i}|} ≃ 1.

Hence, we can assume that α_{p,q} < \frac{π}{4} for all q ∈ U, while keeping the assumption of U being standard (by transfer, as before). As a consequence, the orthogonal projection π_{p}|_{T_{q}} : T_{q} → T_{p} is an isomorphism for any q = (x, g(x)) ∈ M, x ∈ U. Therefore, we can write its inverse π_{p}|_{T_{q}}^{−1}(x + v) = (x + v, g(x) + l_{x}(v)) with some linear map l_{x} : T_{x}T_{p} → T_{g(x)}N_{p} — we want to show that, whenever x ∈ U is near standard g(x + dx) − g(x) − l_{x}(dx) =: ε ≃_{|dx|} 0 for all dx ≃ 0, i.e., that g is continuously differentiable with differential l_{x} = d_{x}g at x, cf.§5.7.9 in [5].

The proof mainly relies on the estimates in Lemma 5. First, let x ∈ U be standard, and dx ≃ 0; denote q := (x, g(x)) and q + dq = (x + dx, g(x + dx)). Then, Lemma 5(ii) implies^{12} that

|g(x + dx) − g(x)| = |dq − (dx, 0)| ≤ tan(α + β) · |dx| ≃ 0

where β ≃ 0 denotes the angle that dq makes with the tangent plane T_{q} at q. Thus, g is continuous. Now, let x ∈ U be near standard, dx, q, and dq as before. Since g is continuous, q = (x, g(x)) is near standard (in M). Consequently, Lemma 5 (i) yields

|g(x + dx) − g(x) − l_{x}(dx)| = |dq − (dx, l_{x}(dx))| ≤

≤ \frac{1}{cos^{2} α_{p,q}(1 − tan α_{p,q} tan β)} tan β · |dx| ≃_{|dx|} 0

as β ≃ 0 and α_{p,q} ≪ \frac{π}{2} . Thus, g is shown to be continuously differentiable, concluding the proof of the theorem. □ Acknowledgements. I would like to thank my office mate E. Tjaden for discussing the linear algebra part of the present paper with me. Also, I would like to thank the referee for pointing out several points where the exposition could be made more readable.

References

[1] U. Hertrich-Jeromin, The surfaces capable of division into Infinitesimal Squares by their Curves of Curvature: A nonstandard analysis approach to classical differential geometry, Math. Intell. 22 (2000), 54-61.

[2] E. Nelson, Internal set theory: A new approach to Nonstandard Analysis, Bull. Amer. Math. Soc. 83 (1977), 1115-1198.

[3] E. Nelson, Unfinished book on Nonstandard Analysis, author’s home page, http://www.math.princeton.edu/˜ nelson/books.html

[4] A.Robert, Nonstandard Analysis, Wiley, New York 1988.

11This is an obvious point where we need that m = dimM ≪ ∞.

12Here, as well as below, we consider q as “0”, dq, (dx, 0), and (dx, l_{x}(dx)) as vectors in the corresponding linear space.

[5] K. Stroyan, W. Luxemburg, Introduction to the Theory of Infinitesimals, Academic Press, New York 1976.

[6] K. Stroyan, Infinitesimal analysis of Curves and Surfaces, in J. Barwise, Handbook of Mathematical Logic, North Holland, Amsterdam 1977.

Dept. Mathematics, MA 8-3 Technical University Berlin Strasse des 17. Juni 136 D-10623 Berlin (Germany) udo@sfb288.math.tu-berlin.de