IMPA - Instituto de Matemática Pura e Aplicada, Rio de Janeiro, RJ, Brazil Introduction to Computational Manifolds and Applications Part 1 - Foundations Prof. Jean Gallier jean@cis.upenn.edu Department of Computer and Information Science University of Pennsylvania Philadelphia, PA, USA
Our definition of manifold is not constructive: it states what a manifold is by assuming that the space already exists. What if we are interested in constructing" a manifold? It turns out that a manifold can be built from what we call a set of gluing data. The idea is to glue open sets in E n in a controlled manner, and then embed them in E d. André Weil introduced this gluing process to define abstract algebraic varieties from irreducible affine sets in a book published in 1946. However, as far as we know, Cindy Grimm and John Hughes were the first to give a constructive definition of manifold. SIGGRAPH, 1995 Computational Manifolds and Applications (CMA) - 2011, IMPA, Rio de Janeiro, RJ, Brazil 2
The pioneering work of Grimm and Hughes allows us to create smooth 2-manifolds (i.e., smooth surfaces equipped with an atlas) in E 3 for the purposes of modeling and simulation. In this lecture we will introduce a formal definition of sets of gluing data, which fixes a problem in the definition given by Grimm and Hughes, and includes a Hausdorff condition. We also introduce the notion of parametric pseudo-manifolds. A parametric pseudo-manifold (PPM) is a topological space defined from a set of gluing data. Under certain conditions (which are often met in practice), PPM s are manifolds in E m. Computational Manifolds and Applications (CMA) - 2011, IMPA, Rio de Janeiro, RJ, Brazil 3
parametric pseudo-manifold θ i (Ω i ) E d M θ j (Ω j ) θ i θ j gluing data ϕ ij Ω i Ω ij Ω ji Ω j E n ϕ ji Computational Manifolds and Applications (CMA) - 2011, IMPA, Rio de Janeiro, RJ, Brazil 4
Let I and K be (possibly infinite) countable sets such that I is nonempty. Definition 7.1. Let n be an integer, with n 1, and k be either an integer, with k 1, or k =. A set of gluing data is a triple, G = (Ω i ) i I, (Ω ij ) (i,j) I I, (ϕ ji ) (i,j) K, satisfying the following properties: Computational Manifolds and Applications (CMA) - 2011, IMPA, Rio de Janeiro, RJ, Brazil 5
(1) For every i I, the set Ω i is a nonempty open subset of E n called parametrization domain, for short, p-domain, and any two distinct p-domains are pairwise disjoint, i.e., Ω i Ω j =, for all i = j. Ω 2 Ω 3 E n... Ω 1 Ω i Computational Manifolds and Applications (CMA) - 2011, IMPA, Rio de Janeiro, RJ, Brazil 6
(2) For every pair (i, j) I I, the set Ω ij is an open subset of Ω i. Furthermore, Ω ii = Ω i and Ω ji = if and only if Ω ij =. Each nonempty subset Ω ij (with i = j) is called a gluing domain. Ω 21 Ω 2 Ω 3 E n Ω 12 Ω 31... Ω 1 Ω i Ω 13 Computational Manifolds and Applications (CMA) - 2011, IMPA, Rio de Janeiro, RJ, Brazil 7
(3) If we let K = {(i, j) I I Ω ij = }, then ϕ ji : Ω ij Ω ji is a C k bijection for every (i, j) K called a transition (or gluing) map. Ω 21 Ω 2 Ω 3 E n ϕ 21 ϕ 12 Ω 12 ϕ 31 ϕ 13 Ω 31... Ω 1 Ω i Ω 13 Computational Manifolds and Applications (CMA) - 2011, IMPA, Rio de Janeiro, RJ, Brazil 8
The transition functions must satisfy the following three conditions: (a) ϕ ii = id Ωi, for all i I, Ω i ϕ ii = id Ωi Computational Manifolds and Applications (CMA) - 2011, IMPA, Rio de Janeiro, RJ, Brazil 9
(b) ϕ ij = ϕ 1 ji, for all (i, j) K, and Ω i ϕ ij Ω j p ϕ 1 ji Computational Manifolds and Applications (CMA) - 2011, IMPA, Rio de Janeiro, RJ, Brazil 10
(c) For all i, j, k, if then Ω ji Ω jk =, ϕ ij (Ω ji Ω jk )=Ω ij Ω ik and ϕ ki (x) =ϕ kj ϕ ji (x), for all x Ω ij Ω ik. Ω ji Ω ik ϕ ij Ω j Ω ji Ω jk Ω i Ω ij Ω jk Ω ij Ω ik Computational Manifolds and Applications (CMA) - 2011, IMPA, Rio de Janeiro, RJ, Brazil 11
ϕ ki (x) =(ϕ kj ϕ ji )(x), for all x (Ω ij Ω ik ). Ω ji Ω ij Ω j ϕ ji Ω i x Ω jk ϕ kj Ω k Ω ki ϕ ki = ϕ kj ϕ ji Ω ik Ω kj Computational Manifolds and Applications (CMA) - 2011, IMPA, Rio de Janeiro, RJ, Brazil 12
The cocycle condition implies conditions (a) and (b): (a) ϕ ii = id Ωi, for all i I, and (b) ϕ ij = ϕ 1 ji, for all (i, j) K. Ω i ϕ ii = id Ωi Ω i ϕ ij Ω j p ϕ 1 ji Computational Manifolds and Applications (CMA) - 2011, IMPA, Rio de Janeiro, RJ, Brazil 13
(4) For every pair (i, j) K, with i = j, for every x (Ω ij ) Ω i and y (Ω ji ) Ω j, there are open balls, V x and V y, centered at x and y, so that no point of V y Ω ji is the image of any point of V x Ω ij by ϕ ji. Ω ij ϕ ji Ω ji V x V y Ω i x ϕ ij y Ω j E n ϕ ji (V x Ω ij ) Computational Manifolds and Applications (CMA) - 2011, IMPA, Rio de Janeiro, RJ, Brazil 14
Given a set of gluing data, G, can we build a manifold from it? The answer is YES! Indeed, such a manifold is built by a quotient construction. Computational Manifolds and Applications (CMA) - 2011, IMPA, Rio de Janeiro, RJ, Brazil 15
The idea is to form the disjoint union, i I Ω i, of the Ω i and then identify Ω ij with Ω ji using ϕ ji. Formally, we define a binary relation,, on i I Ω i as follows: for all x, y i I Ω i, we have x y iff ( (i, j) K)(x Ω ij, y Ω ji, y = ϕ ji (x)). We can prove that is an equivalence relation, which enables us to define the space M G = i I Ω i /. We can also prove that M G is a Hausdorff and second-countable manifold. Computational Manifolds and Applications (CMA) - 2011, IMPA, Rio de Janeiro, RJ, Brazil 16
M G Sketching the proof: [x] For every i I, in i : Ω i i I Ω i is the natural injection. p in 1 p in 3 p in 2 [y] p in 1 Let p : i I Ω i M G be the quotient map, with p in n in 1 (Ω 1 ) in 2 (Ω 2 ) in 3 (Ω 3 ) in n (Ω n ) p(x) =[x]. For every i I, let τ i = p in i : Ω i M G. i I Ω i Let U i = τ i (Ω i ) and ϕ i = τ 1 i. It is immediately verified that (U i, ϕ i ) are charts and that this collection of charts forms a C k atlas for M G. in 1 in 2 in 3 in n x ϕ 21 (x) ϕ 31 (x) y Ω 1 Ω 2 Ω 3 ϕ n1 (y) Ω n Computational Manifolds and Applications (CMA) - 2011, IMPA, Rio de Janeiro, RJ, Brazil 17
Sketching the proof: We now prove that the topology of M G is Hausdorff. Pick [x], [y] M G with [x] = [y], for some x Ω i and some y Ω j. Either τ i (Ω i ) τ j (Ω j )= or τ i (Ω i ) τ j (Ω j ) =. In the former case, as τ i and τ j are homeomorphisms, [x] and [y] belong to the two disjoint open sets τ i (Ω i ) and τ j (Ω j ). In the latter case, we must consider four subcases: Computational Manifolds and Applications (CMA) - 2011, IMPA, Rio de Janeiro, RJ, Brazil 18
Sketching the proof: Ω ij Ω ji (1) x y (2) x y Ω i =Ω j Ω i Ω j Ω ij Ω ji Ω ij Ω ji (3) x y (4) x y Ω i Ω j Ω i Ω j Computational Manifolds and Applications (CMA) - 2011, IMPA, Rio de Janeiro, RJ, Brazil 19
Sketching the proof: (1) If i = j then x and y can be separated by disjoint opens, V x and V y, and as τ i is a homeomorphism, [x] and [y] are separated by the disjoint open subsets τ i (V x ) and τ j (V y ). (1) x y Ω i =Ω j Computational Manifolds and Applications (CMA) - 2011, IMPA, Rio de Janeiro, RJ, Brazil 20
Sketching the proof: (2) If i = j, x Ω i Ω ij and y Ω j Ω ji, then τ i (Ω i Ω ij ) and τ j (Ω j Ω ji ) are disjoint open subsets separating [x] and [y], where Ω ij and Ω ji are the closures of Ω ij and Ω ji, respectively. Ω ij Ω ji (2) x y Ω i Ω j Computational Manifolds and Applications (CMA) - 2011, IMPA, Rio de Janeiro, RJ, Brazil 21
Sketching the proof: (3) If i = j, x Ω ij and y Ω ji, as [x] = [y] and y ϕ ij (y), then x = ϕ ij (y). We can separate x and ϕ ij (y) by disjoint open subsets, V x and V y, and [x] and [y] =[ϕ ij (y)] are separated by the disjoint open subsets τ i (V x ) and τ i (V y ). Ω ij Ω ji (3) x y Ω i Ω j Computational Manifolds and Applications (CMA) - 2011, IMPA, Rio de Janeiro, RJ, Brazil 22
Sketching the proof: (4) If i = j, x (Ω ij ) Ω i and y (Ω ji ) Ω j, then we use condition 4 of Definition 7.1. This condition yields two disjoint open subsets, V x and V y, with x V x and y V y, such that no point of V x Ω ij is equivalent to any point of V y Ω ji, and so τ i (V x ) and τ j (V y ) are disjoint open subsets separating [x] and [y]. Ω ij Ω ji (4) x y Ω i Ω j Computational Manifolds and Applications (CMA) - 2011, IMPA, Rio de Janeiro, RJ, Brazil 23
Sketching the proof: So, the topology of M G is Hausdorff and M G is indeed a manifold. M G is also second-countable (WHY?). Finally, it is trivial to verify that the transition maps of M G are the original gluing functions, ϕ ij, since ϕ i = τ 1 i and ϕ ji = ϕ j ϕ 1 i. Computational Manifolds and Applications (CMA) - 2011, IMPA, Rio de Janeiro, RJ, Brazil 24
Theorem 7.1. For every set of gluing data, G = (Ω i ) i I, (Ω ij ) (i,j) I I, (ϕ ji ) (i,j) K, there is an n-dimensional C k manifold, M G, whose transition maps are the ϕ ji s. Theorem 7.1 is nice, but... Our proof is not constructive; M G is an abstract entity, which may not be orientable, compact, etc. So, we know we can build a manifold from a set of gluing data, but that does not mean we know how to build a "concrete" manifold. For that, we need a formal notion of "concreteness". Computational Manifolds and Applications (CMA) - 2011, IMPA, Rio de Janeiro, RJ, Brazil 25
Parametric Pseudo-Manifolds The notion of "concreteness" is realized as parametric pseudo-manifolds: Definition 7.2. Let n, d, and k be three integers with d > n 1 and k 1 or k =. A parametric C k pseudo-manifold of dimension n in E d (for short, parametric pseudo-manifold or PPM) is a pair, M =(G, (θ i ) i I ), such that G = (Ω i ) i I, (Ω ij ) (i,j) I I, (ϕ ji ) (i,j) K is a set of gluing data, for some finite set I, and each θ i : Ω i E d is C k and satisfies (C) For all (i, j) K, we have θ i = θ j ϕ ji. Computational Manifolds and Applications (CMA) - 2011, IMPA, Rio de Janeiro, RJ, Brazil 26
Manifolds Parametric Pseudo-Manifolds parametric pseudo-manifold θ i (Ω i ) E d M θ j (Ω j ) θ i θ j gluing data ϕ ij Ω i Ω ij Ω ji Ω j E n ϕ ji Computational Manifolds and Applications (CMA) - 2011, IMPA, Rio de Janeiro, RJ, Brazil 27
Parametric Pseudo-Manifolds As usual, we call θ i a parametrization. The subset, M E d, given by M = i I θ i (Ω i ) is called the image of the parametric pseudo-manifold, M. Whenever n = 2 and d = 3, we say that M is a parametric pseudo-surface (or PPS, for short). We also say that M, the image of the PPS M, is a pseudo-surface. Computational Manifolds and Applications (CMA) - 2011, IMPA, Rio de Janeiro, RJ, Brazil 28
Parametric Pseudo-Manifolds Condition C of Definition 7.2, (C) For all (i, j) K, we have θ i = θ j ϕ ji, obviously implies that θ i (Ω ij )=θ j (Ω ji ), for all (i, j) K. Consequently, θ i and θ j are consistent parametrizations of the overlap θ i (Ω ij )=θ j (Ω ji ). Computational Manifolds and Applications (CMA) - 2011, IMPA, Rio de Janeiro, RJ, Brazil 29
Parametric Pseudo-Manifolds θ i (Ω i ) E d M consistent! θ j (Ω j ) θ i θ j ϕ ij Ω i Ω ij Ω ji Ω j E n ϕ ji Computational Manifolds and Applications (CMA) - 2011, IMPA, Rio de Janeiro, RJ, Brazil 30
Parametric Pseudo-Manifolds Thus, the set M, whatever it is, is covered by pieces, U i = θ i (Ω i ), not necessarily open. Each U i is parametrized by θ i, and each overlapping piece, U i U j, is parametrized consistently. The local structure of M is given by the θ i s and its global structure is given by the gluing data. Computational Manifolds and Applications (CMA) - 2011, IMPA, Rio de Janeiro, RJ, Brazil 31
Parametric Pseudo-Manifolds We can equip M with an atlas if we require the θ i s to be injective and to satisfy (C ) For all (i, j) K, (C ) For all (i, j) K, θ i (Ω i ) θ j (Ω j )=θ i (Ω ij )=θ j (Ω ji ). θ i (Ω i ) θ j (Ω j )=. Even if the θ i s are not injective, properties C and C are still desirable since they ensure that θ i (Ω i Ω ij ) and θ j (Ω j Ω ji ) are uniquely parametrized. Unfortunately, properties C and C may be difficult to enforce in practice (at least for surface constructions). Computational Manifolds and Applications (CMA) - 2011, IMPA, Rio de Janeiro, RJ, Brazil 32
Parametric Pseudo-Manifolds Interestingly, regardless whether conditions C and C are satisfied, we can still show that M is the image in E d of the abstract manifold, M G, as stated by Proposition 7.2: Proposition 7.2. Let M =(G, (θ i ) i I ) be a parametric C k pseudo-manifold of dimension n in E d, where G = (Ω i ) i I, (Ω ij ) (i,j) I I, (ϕ ji ) (i,j) K is a set of gluing data, for some finite set I. Then, the parametrization maps, θ i, induce a surjective map, Θ : M G M, from the abstract manifold, M G, specified by G to the image, M E d, of the parametric pseudo-manifold, M, and the following property holds: θ i = Θ τ i, for every Ω i, where τ i : Ω i M G are the parametrization maps of the manifold M G. In particular, every manifold, M E d, such that M is induced by G is the image of M G by a map Θ : M G M. Computational Manifolds and Applications (CMA) - 2011, IMPA, Rio de Janeiro, RJ, Brazil 33
The Evil Cocycle Condition (c) For all i, j, k, if then Ω ji Ω jk =, ϕ ij (Ω ji Ω jk )=Ω ij Ω ik and ϕ ki (x) =ϕ kj ϕ ji (x), for all x Ω ij Ω ik. Ω ji Ω ik ϕ ij Ω j Ω ji Ω jk Ω i Ω ij Ω jk Ω ij Ω ik Computational Manifolds and Applications (CMA) - 2011, IMPA, Rio de Janeiro, RJ, Brazil 34
The Evil Cocycle Condition ϕ ki (x) =(ϕ kj ϕ ji )(x), for all x (Ω ij Ω ik ). Ω ji Ω ij Ω j ϕ ji Ω i x Ω jk ϕ kj Ω k Ω ki ϕ ki = ϕ kj ϕ ji Ω ik Ω kj Computational Manifolds and Applications (CMA) - 2011, IMPA, Rio de Janeiro, RJ, Brazil 35
The statement Parametric Pseudo-Manifolds The Evil Cocycle Condition if Ω ji Ω jk = then ϕ ij (Ω ji Ω jk )=Ω ij Ω ik is necessary for guaranteeing the transitivity of the equivalence relation. Ω ji Ω ik ϕ ij Ω j Ω ji Ω jk Ω i Ω ij Ω jk Ω ij Ω ik Computational Manifolds and Applications (CMA) - 2011, IMPA, Rio de Janeiro, RJ, Brazil 36
The Evil Cocycle Condition Consider the p-domains (i.e., open line intervals) Ω 1 =]0, 3 [, Ω 2 =]4, 5 [, and Ω 3 =]6, 9 [. Ω 1 Ω 2 Ω 3 0 1 2 3 4 5 6 7 8 9 E Consider the gluing domains Ω 12 =]0, 1 [ Ω 13 =]2, 3 [, Ω 21 = Ω 23 =]4, 5 [, Ω 32 =]8, 9 [ Ω 31 =]6, 7 [. Ω 12 Ω 13 Ω 21 = Ω 23 Ω 31 Ω 32 0 1 2 3 4 5 6 7 8 9 E Computational Manifolds and Applications (CMA) - 2011, IMPA, Rio de Janeiro, RJ, Brazil 37
The Evil Cocycle Condition Consider the transition maps: ϕ 21 (x) =x + 4, ϕ 32 (x) =x + 4 and ϕ 31 (x) =x + 4. Ω 12 Ω 13 Ω 21 = Ω 23 Ω 31 Ω 32 0 1 2 3 4 5 6 7 8 9 E ϕ 21 ϕ 31 ϕ 32 Computational Manifolds and Applications (CMA) - 2011, IMPA, Rio de Janeiro, RJ, Brazil 38
The Evil Cocycle Condition Obviously, ϕ32 ϕ 21 (x) =x + 8, for all x Ω12. Ω 12 Ω 13 Ω 21 = Ω 23 Ω 31 Ω 32 0 1 2 3 4 5 6 7 8 9 E ϕ 21 ϕ 32 ϕ 21 (0.5) =4.5 and ϕ 32 (4.5) =8.5 = 0.5 4.5 and 4.5 8.5 So, if were transitive, then we would have 0.5 8.5. But... Computational Manifolds and Applications (CMA) - 2011, IMPA, Rio de Janeiro, RJ, Brazil 39
The Evil Cocycle Condition it turns out that ϕ 31 is undefined at 0.5. Ω 12 Ω 13 Ω 21 = Ω 23 Ω 31 Ω 32 0 1 2 3 4 5 6 7 8 9 E ϕ 31 So, 0.5 8.5. The reason is that ϕ 31 and ϕ 32 ϕ 21 have disjoint domains. Computational Manifolds and Applications (CMA) - 2011, IMPA, Rio de Janeiro, RJ, Brazil 40
The Evil Cocycle Condition The reason they have disjoint domains is that condition "c" is not satisfied: if Ω 21 Ω 23 = then ϕ 12 (Ω 21 Ω 23 )=Ω 12 Ω 13. Ω 12 Ω 13 Ω 21 = Ω 23 Ω 31 Ω 32 0 1 2 3 4 5 6 7 8 9 E Indeed but Ω 21 Ω 23 = Ω 2 =]4, 5 [ =, ϕ 12 (Ω 21 Ω 23 )=]0, 1 [ = = Ω 12 Ω 13. Computational Manifolds and Applications (CMA) - 2011, IMPA, Rio de Janeiro, RJ, Brazil 41