Lecture 9: the Whitney Embedding Theorem
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Planar Embeddings of Minc's Continuum and Generalizations
PLANAR EMBEDDINGS OF MINC’S CONTINUUM AND GENERALIZATIONS ANA ANUSIˇ C´ Abstract. We show that if f : I → I is piecewise monotone, post-critically finite, x X I,f and locally eventually onto, then for every point ∈ =←− lim( ) there exists a planar embedding of X such that x is accessible. In particular, every point x in Minc’s continuum XM from [11, Question 19 p. 335] can be embedded accessibly. All constructed embeddings are thin, i.e., can be covered by an arbitrary small chain of open sets which are connected in the plane. 1. Introduction The main motivation for this study is the following long-standing open problem: Problem (Nadler and Quinn 1972 [20, p. 229] and [21]). Let X be a chainable contin- uum, and x ∈ X. Is there a planar embedding of X such that x is accessible? The importance of this problem is illustrated by the fact that it appears at three independent places in the collection of open problems in Continuum Theory published in 2018 [10, see Question 1, Question 49, and Question 51]. We will give a positive answer to the Nadler-Quinn problem for every point in a wide class of chainable continua, which includes←− lim(I, f) for a simplicial locally eventually onto map f, and in particular continuum XM introduced by Piotr Minc in [11, Question 19 p. 335]. Continuum XM was suspected to have a point which is inaccessible in every planar embedding of XM . A continuum is a non-empty, compact, connected, metric space, and it is chainable if arXiv:2010.02969v1 [math.GN] 6 Oct 2020 it can be represented as an inverse limit with bonding maps fi : I → I, i ∈ N, which can be assumed to be onto and piecewise linear. -
Ricci Curvature and Minimal Submanifolds
Pacific Journal of Mathematics RICCI CURVATURE AND MINIMAL SUBMANIFOLDS Thomas Hasanis and Theodoros Vlachos Volume 197 No. 1 January 2001 PACIFIC JOURNAL OF MATHEMATICS Vol. 197, No. 1, 2001 RICCI CURVATURE AND MINIMAL SUBMANIFOLDS Thomas Hasanis and Theodoros Vlachos The aim of this paper is to find necessary conditions for a given complete Riemannian manifold to be realizable as a minimal submanifold of a unit sphere. 1. Introduction. The general question that served as the starting point for this paper was to find necessary conditions on those Riemannian metrics that arise as the induced metrics on minimal hypersurfaces or submanifolds of hyperspheres of a Euclidean space. There is an abundance of complete minimal hypersurfaces in the unit hy- persphere Sn+1. We recall some well known examples. Let Sm(r) = {x ∈ Rn+1, |x| = r},Sn−m(s) = {y ∈ Rn−m+1, |y| = s}, where r and s are posi- tive numbers with r2 + s2 = 1; then Sm(r) × Sn−m(s) = {(x, y) ∈ Rn+2, x ∈ Sm(r), y ∈ Sn−m(s)} is a hypersurface of the unit hypersphere in Rn+2. As is well known, this hypersurface has two distinct constant principal cur- vatures: One is s/r of multiplicity m, the other is −r/s of multiplicity n − m. This hypersurface is called a Clifford hypersurface. Moreover, it is minimal only in the case r = pm/n, s = p(n − m)/n and is called a Clifford minimal hypersurface. Otsuki [11] proved that if M n is a com- pact minimal hypersurface in Sn+1 with two distinct principal curvatures of multiplicity greater than 1, then M n is a Clifford minimal hypersur- face Sm(pm/n) × Sn−m(p(n − m)/n), 1 < m < n − 1. -
Neural Subgraph Matching
Neural Subgraph Matching NEURAL SUBGRAPH MATCHING Rex Ying, Andrew Wang, Jiaxuan You, Chengtao Wen, Arquimedes Canedo, Jure Leskovec Stanford University and Siemens Corporate Technology ABSTRACT Subgraph matching is the problem of determining the presence of a given query graph in a large target graph. Despite being an NP-complete problem, the subgraph matching problem is crucial in domains ranging from network science and database systems to biochemistry and cognitive science. However, existing techniques based on combinatorial matching and integer programming cannot handle matching problems with both large target and query graphs. Here we propose NeuroMatch, an accurate, efficient, and robust neural approach to subgraph matching. NeuroMatch decomposes query and target graphs into small subgraphs and embeds them using graph neural networks. Trained to capture geometric constraints corresponding to subgraph relations, NeuroMatch then efficiently performs subgraph matching directly in the embedding space. Experiments demonstrate that NeuroMatch is 100x faster than existing combinatorial approaches and 18% more accurate than existing approximate subgraph matching methods. 1.I NTRODUCTION Given a query graph, the problem of subgraph isomorphism matching is to determine if a query graph is isomorphic to a subgraph of a large target graph. If the graphs include node and edge features, both the topology as well as the features should be matched. Subgraph matching is a crucial problem in many biology, social network and knowledge graph applications (Gentner, 1983; Raymond et al., 2002; Yang & Sze, 2007; Dai et al., 2019). For example, in social networks and biomedical network science, researchers investigate important subgraphs by counting them in a given network (Alon et al., 2008). -
Isoparametric Hypersurfaces with Four Principal Curvatures Revisited
ISOPARAMETRIC HYPERSURFACES WITH FOUR PRINCIPAL CURVATURES REVISITED QUO-SHIN CHI Abstract. The classification of isoparametric hypersurfaces with four principal curvatures in spheres in [2] hinges on a crucial charac- terization, in terms of four sets of equations of the 2nd fundamental form tensors of a focal submanifold, of an isoparametric hypersur- face of the type constructed by Ferus, Karcher and Munzner.¨ The proof of the characterization in [2] is an extremely long calculation by exterior derivatives with remarkable cancellations, which is mo- tivated by the idea that an isoparametric hypersurface is defined by an over-determined system of partial differential equations. There- fore, exterior differentiating sufficiently many times should gather us enough information for the conclusion. In spite of its elemen- tary nature, the magnitude of the calculation and the surprisingly pleasant cancellations make it desirable to understand the under- lying geometric principles. In this paper, we give a conceptual, and considerably shorter, proof of the characterization based on Ozeki and Takeuchi's expan- sion formula for the Cartan-Munzner¨ polynomial. Along the way the geometric meaning of these four sets of equations also becomes clear. 1. Introduction In [2], isoparametric hypersurfaces with four principal curvatures and multiplicities (m1; m2); m2 ≥ 2m1 − 1; in spheres were classified to be exactly the isoparametric hypersurfaces of F KM-type constructed by Ferus Karcher and Munzner¨ [4]. The classification goes as follows. Let M+ be a focal submanifold of codimension m1 of an isoparametric hypersurface in a sphere, and let N be the normal bundle of M+ in the sphere. Suppose on the unit normal bundle UN of N there hold true 1991 Mathematics Subject Classification. -
Tight Immersions of Simplicial Surfaces in Three Space
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Pergamon Topology Vol. 35, No. 4, pp. 863-873, 19% Copyright Q 1996 Elsevier Sctence Ltd Printed in Great Britain. All tights reserved CO4&9383/96 $15.00 + 0.00 0040-9383(95)00047-X TIGHT IMMERSIONS OF SIMPLICIAL SURFACES IN THREE SPACE DAVIDE P. CERVONE (Received for publication 13 July 1995) 1. INTRODUCTION IN 1960,Kuiper [12,13] initiated the study of tight immersions of surfaces in three space, and showed that every compact surface admits a tight immersion except for the Klein bottle, the real projective plane, and possibly the real projective plane with one handle. The fate of the latter surface went unresolved for 30 years until Haab recently proved [9] that there is no smooth tight immersion of the projective plane with one handle. In light of this, a recently described simplicial tight immersion of this surface [6] came as quite a surprise, and its discovery completes Kuiper’s initial survey. Pinkall [17] broadened the question in 1985 by looking for tight immersions in each equivalence class of immersed surfaces under image homotopy. He first produced a complete description of these classes, and proceeded to generate tight examples in all but finitely many of them. In this paper, we improve Pinkall’s result by giving tight simplicial examples in all but two of the immersion classes under image homotopy for which tight immersions are possible, and in particular, we point out and resolve an error in one of his examples that would have left an infinite number of immersion classes with no tight examples. -
Lecture Notes on Foliation Theory
INDIAN INSTITUTE OF TECHNOLOGY BOMBAY Department of Mathematics Seminar Lectures on Foliation Theory 1 : FALL 2008 Lecture 1 Basic requirements for this Seminar Series: Familiarity with the notion of differential manifold, submersion, vector bundles. 1 Some Examples Let us begin with some examples: m d m−d (1) Write R = R × R . As we know this is one of the several cartesian product m decomposition of R . Via the second projection, this can also be thought of as a ‘trivial m−d vector bundle’ of rank d over R . This also gives the trivial example of a codim. d- n d foliation of R , as a decomposition into d-dimensional leaves R × {y} as y varies over m−d R . (2) A little more generally, we may consider any two manifolds M, N and a submersion f : M → N. Here M can be written as a disjoint union of fibres of f each one is a submanifold of dimension equal to dim M − dim N = d. We say f is a submersion of M of codimension d. The manifold structure for the fibres comes from an atlas for M via the surjective form of implicit function theorem since dfp : TpM → Tf(p)N is surjective at every point of M. We would like to consider this description also as a codim d foliation. However, this is also too simple minded one and hence we would call them simple foliations. If the fibres of the submersion are connected as well, then we call it strictly simple. (3) Kronecker Foliation of a Torus Let us now consider something non trivial. -
Lecture 10: Tubular Neighborhood Theorem
LECTURE 10: TUBULAR NEIGHBORHOOD THEOREM 1. Generalized Inverse Function Theorem We will start with Theorem 1.1 (Generalized Inverse Function Theorem, compact version). Let f : M ! N be a smooth map that is one-to-one on a compact submanifold X of M. Moreover, suppose dfx : TxM ! Tf(x)N is a linear diffeomorphism for each x 2 X. Then f maps a neighborhood U of X in M diffeomorphically onto a neighborhood V of f(X) in N. Note that if we take X = fxg, i.e. a one-point set, then is the inverse function theorem we discussed in Lecture 2 and Lecture 6. According to that theorem, one can easily construct neighborhoods U of X and V of f(X) so that f is a local diffeomor- phism from U to V . To get a global diffeomorphism from a local diffeomorphism, we will need the following useful lemma: Lemma 1.2. Suppose f : M ! N is a local diffeomorphism near every x 2 M. If f is invertible, then f is a global diffeomorphism. Proof. We need to show f −1 is smooth. Fix any y = f(x). The smoothness of f −1 at y depends only on the behaviour of f −1 near y. Since f is a diffeomorphism from a −1 neighborhood of x onto a neighborhood of y, f is smooth at y. Proof of Generalized IFT, compact version. According to Lemma 1.2, it is enough to prove that f is one-to-one in a neighborhood of X. We may embed M into RK , and consider the \"-neighborhood" of X in M: X" = fx 2 M j d(x; X) < "g; where d(·; ·) is the Euclidean distance. -
Some Planar Embeddings of Chainable Continua Can Be
Some planar embeddings of chainable continua can be expressed as inverse limit spaces by Susan Pamela Schwartz A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Montana State University © Copyright by Susan Pamela Schwartz (1992) Abstract: It is well known that chainable continua can be expressed as inverse limit spaces and that chainable continua are embeddable in the plane. We give necessary and sufficient conditions for the planar embeddings of chainable continua to be realized as inverse limit spaces. As an example, we consider the Knaster continuum. It has been shown that this continuum can be embedded in the plane in such a manner that any given composant is accessible. We give inverse limit expressions for embeddings of the Knaster continuum in which the accessible composant is specified. We then show that there are uncountably many non-equivalent inverse limit embeddings of this continuum. SOME PLANAR EMBEDDINGS OF CHAIN ABLE OONTINUA CAN BE EXPRESSED AS INVERSE LIMIT SPACES by Susan Pamela Schwartz A thesis submitted in partial fulfillment of the requirements for the degree of . Doctor of Philosophy in Mathematics MONTANA STATE UNIVERSITY Bozeman, Montana February 1992 D 3 l% ii APPROVAL of a thesis submitted by Susan Pamela Schwartz This thesis has been read by each member of the thesis committee and has been found to be satisfactory regarding content, English usage, format, citations, bibliographic style, and consistency, and is ready for submission to the College of Graduate Studies. g / / f / f z Date Chairperson, Graduate committee Approved for the Major Department ___ 2 -J2 0 / 9 Date Head, Major Department Approved for the College of Graduate Studies Date Graduate Dean iii STATEMENT OF PERMISSION TO USE . -
Cauchy Graph Embedding
Cauchy Graph Embedding Dijun Luo [email protected] Chris Ding [email protected] Feiping Nie [email protected] Heng Huang [email protected] The University of Texas at Arlington, 701 S. Nedderman Drive, Arlington, TX 76019 Abstract classify unsupervised embedding approaches into two cat- Laplacian embedding provides a low- egories. Approaches in the first category are to embed data dimensional representation for the nodes of into a linear space via linear transformations, such as prin- a graph where the edge weights denote pair- ciple component analysis (PCA) (Jolliffe, 2002) and mul- wise similarity among the node objects. It is tidimensional scaling (MDS) (Cox & Cox, 2001). Both commonly assumed that the Laplacian embed- PCA and MDS are eigenvector methods and can model lin- ding results preserve the local topology of the ear variabilities in high-dimensional data. They have been original data on the low-dimensional projected long known and widely used in many machine learning ap- subspaces, i.e., for any pair of graph nodes plications. with large similarity, they should be embedded However, the underlying structure of real data is often closely in the embedded space. However, in highly nonlinear and hence cannot be accurately approx- this paper, we will show that the Laplacian imated by linear manifolds. The second category ap- embedding often cannot preserve local topology proaches embed data in a nonlinear manner based on differ- well as we expected. To enhance the local topol- ent purposes. Recently several promising nonlinear meth- ogy preserving property in graph embedding, ods have been proposed, including IsoMAP (Tenenbaum we propose a novel Cauchy graph embedding et al., 2000), Local Linear Embedding (LLE) (Roweis & which preserves the similarity relationships of Saul, 2000), Local Tangent Space Alignment (Zhang & the original data in the embedded space via a Zha, 2004), Laplacian Embedding/Eigenmap (Hall, 1971; new objective. -
Two Classes of Slant Surfaces in Nearly Kahler Six Sphere
TWO CLASSES OF SLANT SURFACES IN NEARLY KAHLER¨ SIX SPHERE K. OBRENOVIC´ AND S. VUKMIROVIC´ Abstract. In this paper we find examples of slant surfaces in the nearly K¨ahler six sphere. First, we characterize two-dimensional small and great spheres which are slant. Their description is given in terms of the associative 3-form in Im O. Later on, we classify the slant surfaces of S6 which are orbits of maximal torus in G2. We show that these orbits are flat tori which are linearly S5 S6 1 π . full in ⊂ and that their slant angle is between arccos 3 and 2 Among them we find one parameter family of minimal orbits. 1. Introduction It is known that S2 and S6 are the only spheres that admit an almost complex structure. The best known Hermitian almost complex structure J on S6 is defined using octonionic multiplication. It not integrable, but satisfies condition ( X J)X = 0, for the Levi-Civita connection and every vector field X on S6. Therefore,∇ sphere S6 with this structure J is usually∇ referred as nearly K¨ahler six sphere. Submanifolds of nearly Kahler¨ sphere S6 are subject of intensive research. A. Gray [7] proved that almost complex submanifolds of nearly K¨ahler S6 are necessarily two-dimensional and minimal. In paper [3] Bryant showed that any Riemannian surface can be embedded in the six sphere as an almost complex submanifold. Almost complex surfaces were further investigated in paper [1] and classified into four types. Totally real submanifolds of S6 can be of dimension two or three. -
MA4E0 Lie Groups
MA4E0 Lie Groups December 5, 2012 Contents 0 Topological Groups 1 1 Manifolds 2 2 Lie Groups 3 3 The Lie Algebra 6 4 The Tangent space 8 5 The Lie Algebra of a Lie Group 10 6 Integrating Vector fields and the exponential map 13 7 Lie Subgroups 17 8 Continuous homomorphisms are smooth 22 9 Distributions and Frobenius Theorem 23 10 Lie Group topics 24 These notes are based on the 2012 MA4E0 Lie Groups course, taught by John Rawnsley, typeset by Matthew Egginton. No guarantee is given that they are accurate or applicable, but hopefully they will assist your study. Please report any errors, factual or typographical, to [email protected] i MA4E0 Lie Groups Lecture Notes Autumn 2012 0 Topological Groups Let G be a group and then write the group structure in terms of maps; multiplication becomes m : G × G ! G −1 defined by m(g1; g2) = g1g2 and inversion becomes i : G ! G defined by i(g) = g . If we suppose that there is a topology on G as a set given by a subset T ⊂ P (G) with the usual rules. We give G × G the product topology. Then we require that m and i are continuous maps. Then G with this topology is a topological group Examples include 1. G any group with the discrete topology 2. Rn with the Euclidean topology and m(x; y) = x + y and i(x) = −x 1 2 2 2 3. S = f(x; y) 2 R : x + y = 1g = fz 2 C : jzj = 1g with m(z1; z2) = z1z2 and i(z) =z ¯. -
Strong Inverse Limit Reflection
Strong Inverse Limit Reflection Scott Cramer March 4, 2016 Abstract We show that the axiom Strong Inverse Limit Reflection holds in L(Vλ+1) assuming the large cardinal axiom I0. This reflection theorem both extends results of [4], [5], and [3], and has structural implications for L(Vλ+1), as described in [3]. Furthermore, these results together highlight an analogy between Strong Inverse Limit Reflection and the Axiom of Determinacy insofar as both act as fundamental regularity properties. The study of L(Vλ+1) was initiated by H. Woodin in order to prove properties of L(R) under large cardinal assumptions. In particular he showed that L(R) satisfies the Axiom of Determinacy (AD) if there exists a non-trivial elementary embedding j : L(Vλ+1) ! L(Vλ+1) with crit (j) < λ (an axiom called I0). We investigate an axiom called Strong Inverse Limit Reflection for L(Vλ+1) which is in some sense analogous to AD for L(R). Our main result is to show that if I0 holds at λ then Strong Inverse Limit Reflection holds in L(Vλ+1). Strong Inverse Limit Reflection is a strong form of a reflection property for inverse limits. Axioms of this form generally assert the existence of a collection of embeddings reflecting a certain amount of L(Vλ+1), together with a largeness assumption on the collection. There are potentially many different types of axioms of this form which could be considered, but we concentrate on a particular form which, by results in [3], has certain structural consequences for L(Vλ+1), such as a version of the perfect set property.