On Manifolds of Negative Curvature, Geodesic Flow, and Ergodicity
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A Geometric Take on Metric Learning
A Geometric take on Metric Learning Søren Hauberg Oren Freifeld Michael J. Black MPI for Intelligent Systems Brown University MPI for Intelligent Systems Tubingen,¨ Germany Providence, US Tubingen,¨ Germany [email protected] [email protected] [email protected] Abstract Multi-metric learning techniques learn local metric tensors in different parts of a feature space. With such an approach, even simple classifiers can be competitive with the state-of-the-art because the distance measure locally adapts to the struc- ture of the data. The learned distance measure is, however, non-metric, which has prevented multi-metric learning from generalizing to tasks such as dimensional- ity reduction and regression in a principled way. We prove that, with appropriate changes, multi-metric learning corresponds to learning the structure of a Rieman- nian manifold. We then show that this structure gives us a principled way to perform dimensionality reduction and regression according to the learned metrics. Algorithmically, we provide the first practical algorithm for computing geodesics according to the learned metrics, as well as algorithms for computing exponential and logarithmic maps on the Riemannian manifold. Together, these tools let many Euclidean algorithms take advantage of multi-metric learning. We illustrate the approach on regression and dimensionality reduction tasks that involve predicting measurements of the human body from shape data. 1 Learning and Computing Distances Statistics relies on measuring distances. When the Euclidean metric is insufficient, as is the case in many real problems, standard methods break down. This is a key motivation behind metric learning, which strives to learn good distance measures from data. -
Connections on Bundles Md
Dhaka Univ. J. Sci. 60(2): 191-195, 2012 (July) Connections on Bundles Md. Showkat Ali, Md. Mirazul Islam, Farzana Nasrin, Md. Abu Hanif Sarkar and Tanzia Zerin Khan Department of Mathematics, University of Dhaka, Dhaka 1000, Bangladesh, Email: [email protected] Received on 25. 05. 2011.Accepted for Publication on 15. 12. 2011 Abstract This paper is a survey of the basic theory of connection on bundles. A connection on tangent bundle , is called an affine connection on an -dimensional smooth manifold . By the general discussion of affine connection on vector bundles that necessarily exists on which is compatible with tensors. I. Introduction = < , > (2) In order to differentiate sections of a vector bundle [5] or where <, > represents the pairing between and ∗. vector fields on a manifold we need to introduce a Then is a section of , called the absolute differential structure called the connection on a vector bundle. For quotient or the covariant derivative of the section along . example, an affine connection is a structure attached to a differentiable manifold so that we can differentiate its Theorem 1. A connection always exists on a vector bundle. tensor fields. We first introduce the general theorem of Proof. Choose a coordinate covering { }∈ of . Since connections on vector bundles. Then we study the tangent vector bundles are trivial locally, we may assume that there is bundle. is a -dimensional vector bundle determine local frame field for any . By the local structure of intrinsically by the differentiable structure [8] of an - connections, we need only construct a × matrix on dimensional smooth manifold . each such that the matrices satisfy II. -
Characterization of Exact Lumpability for Vector Fields on Smooth Manifolds
Preprint. Final version in: Differ.Geom.Appl. 48 (2016) 46-60 doi: 10.1016/j.difgeo.2016.06.001 Characterization of Exact Lumpability for Vector Fields on Smooth Manifolds Leonhard Horstmeyer∗ † Fatihcan M. Atay‡ Abstract We characterize the exact lumpability of smooth vector fields on smooth man- ifolds. We derive necessary and sufficient conditions for lumpability and express them from four different perspectives, thus simplifying and generalizing various re- sults from the literature that exist for Euclidean spaces. We introduce a partial connection on the pullback bundle that is related to the Bott connection and be- haves like a Lie derivative. The lumping conditions are formulated in terms of the differential of the lumping map, its covariant derivative with respect to the con- nection and their respective kernels. Some examples are discussed to illustrate the theory. PACS numbers: 02.40.-k, 02.30.Hq, 02.40.Hw AMS classification scheme numbers: 37C10, 34C40, 58A30, 53B05, 34A05 Keywords: lumping, aggregation, dimensional reduction, Bott connection 1 Introduction Dimensional reduction is an important aspect in the study of smooth dynamical systems and in particular in modeling with ordinary differential equations (ODEs). Often a reduction can elucidate key mechanisms, find decoupled subsystems, reveal conserved quantities, make the problem computationally tractable, or rid it from redundancies. A dimensional reduction by which micro state variables are aggregated into macro state variables also goes by the name of lumping. Starting from a micro state dynamics, this arXiv:1607.01237v2 [math.DG] 6 Jul 2016 aggregation induces a lumped dynamics on the macro state space. Whenever a non- trivial lumping, one that is neither the identity nor maps to a single point, confers the ∗Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103 Leipzig, Germany. -
FOLIATIONS Introduction. the Study of Foliations on Manifolds Has a Long
BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 80, Number 3, May 1974 FOLIATIONS BY H. BLAINE LAWSON, JR.1 TABLE OF CONTENTS 1. Definitions and general examples. 2. Foliations of dimension-one. 3. Higher dimensional foliations; integrability criteria. 4. Foliations of codimension-one; existence theorems. 5. Notions of equivalence; foliated cobordism groups. 6. The general theory; classifying spaces and characteristic classes for foliations. 7. Results on open manifolds; the classification theory of Gromov-Haefliger-Phillips. 8. Results on closed manifolds; questions of compact leaves and stability. Introduction. The study of foliations on manifolds has a long history in mathematics, even though it did not emerge as a distinct field until the appearance in the 1940's of the work of Ehresmann and Reeb. Since that time, the subject has enjoyed a rapid development, and, at the moment, it is the focus of a great deal of research activity. The purpose of this article is to provide an introduction to the subject and present a picture of the field as it is currently evolving. The treatment will by no means be exhaustive. My original objective was merely to summarize some recent developments in the specialized study of codimension-one foliations on compact manifolds. However, somewhere in the writing I succumbed to the temptation to continue on to interesting, related topics. The end product is essentially a general survey of new results in the field with, of course, the customary bias for areas of personal interest to the author. Since such articles are not written for the specialist, I have spent some time in introducing and motivating the subject. -
INTRODUCTION to ALGEBRAIC GEOMETRY 1. Preliminary Of
INTRODUCTION TO ALGEBRAIC GEOMETRY WEI-PING LI 1. Preliminary of Calculus on Manifolds 1.1. Tangent Vectors. What are tangent vectors we encounter in Calculus? 2 0 (1) Given a parametrised curve α(t) = x(t); y(t) in R , α (t) = x0(t); y0(t) is a tangent vector of the curve. (2) Given a surface given by a parameterisation x(u; v) = x(u; v); y(u; v); z(u; v); @x @x n = × is a normal vector of the surface. Any vector @u @v perpendicular to n is a tangent vector of the surface at the corresponding point. (3) Let v = (a; b; c) be a unit tangent vector of R3 at a point p 2 R3, f(x; y; z) be a differentiable function in an open neighbourhood of p, we can have the directional derivative of f in the direction v: @f @f @f D f = a (p) + b (p) + c (p): (1.1) v @x @y @z In fact, given any tangent vector v = (a; b; c), not necessarily a unit vector, we still can define an operator on the set of functions which are differentiable in open neighbourhood of p as in (1.1) Thus we can take the viewpoint that each tangent vector of R3 at p is an operator on the set of differential functions at p, i.e. @ @ @ v = (a; b; v) ! a + b + c j ; @x @y @z p or simply @ @ @ v = (a; b; c) ! a + b + c (1.2) @x @y @z 3 with the evaluation at p understood. -
3. Tensor Fields
3.1 The tangent bundle. So far we have considered vectors and tensors at a point. We now wish to consider fields of vectors and tensors. The union of all tangent spaces is called the tangent bundle and denoted TM: TM = ∪x∈M TxM . (3.1.1) The tangent bundle can be given a natural manifold structure derived from the manifold structure of M. Let π : TM → M be the natural projection that associates a vector v ∈ TxM to the point x that it is attached to. Let (UA, ΦA) be an atlas on M. We construct an atlas on TM as follows. The domain of a −1 chart is π (UA) = ∪x∈UA TxM, i.e. it consists of all vectors attached to points that belong to UA. The local coordinates of a vector v are (x1,...,xn, v1,...,vn) where (x1,...,xn) are the coordinates of x and v1,...,vn are the components of the vector with respect to the coordinate basis (as in (2.2.7)). One can easily check that a smooth coordinate trasformation on M induces a smooth coordinate transformation on TM (the transformation of the vector components is given by (2.3.10), so if M is of class Cr, TM is of class Cr−1). In a similar way one defines the cotangent bundle ∗ ∗ T M = ∪x∈M Tx M , (3.1.2) the tensor bundles s s TMr = ∪x∈M TxMr (3.1.3) and the bundle of p-forms p p Λ M = ∪x∈M Λx . (3.1.4) 3.2 Vector and tensor fields. -
Lecture 2 Tangent Space, Differential Forms, Riemannian Manifolds
Lecture 2 tangent space, differential forms, Riemannian manifolds differentiable manifolds A manifold is a set that locally look like Rn. For example, a two-dimensional sphere S2 can be covered by two subspaces, one can be the northen hemisphere extended slightly below the equator and another can be the southern hemisphere extended slightly above the equator. Each patch can be mapped smoothly into an open set of R2. In general, a manifold M consists of a family of open sets Ui which covers M, i.e. iUi = M, n ∪ and, for each Ui, there is a continuous invertible map ϕi : Ui R . To be precise, to define → what we mean by a continuous map, we has to define M as a topological space first. This requires a certain set of properties for open sets of M. We will discuss this in a couple of weeks. For now, we assume we know what continuous maps mean for M. If you need to know now, look at one of the standard textbooks (e.g., Nakahara). Each (Ui, ϕi) is called a coordinate chart. Their collection (Ui, ϕi) is called an atlas. { } The map has to be one-to-one, so that there is an inverse map from the image ϕi(Ui) to −1 Ui. If Ui and Uj intersects, we can define a map ϕi ϕj from ϕj(Ui Uj)) to ϕi(Ui Uj). ◦ n ∩ ∩ Since ϕj(Ui Uj)) to ϕi(Ui Uj) are both subspaces of R , we express the map in terms of n ∩ ∩ functions and ask if they are differentiable. -
5. the Inverse Function Theorem We Now Want to Aim for a Version of the Inverse Function Theorem
5. The inverse function theorem We now want to aim for a version of the Inverse function Theorem. In differential geometry, the inverse function theorem states that if a function is an isomorphism on tangent spaces, then it is locally an isomorphism. Unfortunately this is too much to expect in algebraic geometry, since the Zariski topology is too weak for this to be true. For example consider a curve which double covers another curve. At any point where there are two points in the fibre, the map on tangent spaces is an isomorphism. But there is no Zariski neighbourhood of any point where the map is an isomorphism. Thus a minimal requirement is that the morphism is a bijection. Note that this is not enough in general for a morphism between al- gebraic varieties to be an isomorphism. For example in characteristic p, Frobenius is nowhere smooth and even in characteristic zero, the parametrisation of the cuspidal cubic is a bijection but not an isomor- phism. Lemma 5.1. If f : X −! Y is a projective morphism with finite fibres, then f is finite. Proof. Since the result is local on the base, we may assume that Y is affine. By assumption X ⊂ Y × Pn and we are projecting onto the first factor. Possibly passing to a smaller open subset of Y , we may assume that there is a point p 2 Pn such that X does not intersect Y × fpg. As the blow up of Pn at p, fibres over Pn−1 with fibres isomorphic to P1, and the composition of finite morphisms is finite, we may assume that n = 1, by induction on n. -
Lorentzian Cobordisms, Compact Horizons and the Generic Condition
Lorentzian Cobordisms, Compact Horizons and the Generic Condition Eric Larsson Master of Science Thesis Stockholm, Sweden 2014 Lorentzian Cobordisms, Compact Horizons and the Generic Condition Eric Larsson Master’s Thesis in Mathematics (30 ECTS credits) Degree programme in Engineering Physics (300 credits) Royal Institute of Technology year 2014 Supervisor at KTH was Mattias Dahl Examiner was Mattias Dahl TRITA-MAT-E 2014:29 ISRN-KTH/MAT/E--14/29--SE Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci iii Abstract We consider the problem of determining which conditions are necessary for cobordisms to admit Lorentzian metrics with certain properties. In particu- lar, we prove a result originally due to Tipler without a smoothness hypothe- sis necessary in the original proof. In doing this, we prove that compact hori- zons in a smooth spacetime satisfying the null energy condition are smooth. We also prove that the ”generic condition” is indeed generic in the set of Lorentzian metrics on a given manifold. Acknowledgements I would like to thank my advisor Mattias Dahl for invaluable advice and en- couragement. Thanks also to Hans Ringström. Special thanks to Marc Nardmann for feedback on Chapter 2. Contents Contents iv 1 Lorentzian cobordisms 1 1.1 Existence of Lorentzian cobordisms . 2 1.2 Lorentzian cobordisms and causality . 4 1.3 Lorentzian cobordisms and energy conditions . 8 1.3.1 C 2 null hypersurfaces . 9 1.3.1.1 The null Weingarten map . 9 1.3.1.2 Generator flow on C 2 null hypersurfaces . -
The Holomorphic Φ-Sectional Curvature of Tangent Sphere Bundles with Sasakian Structures
ANALELE S¸TIINT¸IFICE ALE UNIVERSITAT¸II˘ \AL.I. CUZA" DIN IAS¸I (S.N.) MATEMATICA,˘ Tomul LVII, 2011, Supliment DOI: 10.2478/v10157-011-0004-5 THE HOLOMORPHIC '-SECTIONAL CURVATURE OF TANGENT SPHERE BUNDLES WITH SASAKIAN STRUCTURES BY S. L. DRUT¸ A-ROMANIUC˘ and V. OPROIU Abstract. We study the holomorphic '-sectional curvature of the natural diagonal tangent sphere bundles with Sasakian structures, determined by Drut¸a-Romaniuc˘ and Oproiu. After finding the explicit expressions for the components of the curvature tensor field and of the curvature tensor field corresponding to the Sasakian space forms, we find that there are no tangent sphere bundles of natural diagonal lift type of constant holomorphic '-sectional curvature. Mathematics Subject Classification 2000: 53C05, 53C15, 53C55. Key words: natural lift, tangent sphere bundle, Sasakian structure, holomorphic '-sectional curvature. 1. Introduction The geometry of the tangent sphere bundles of constant radius r, endowed with the metrics induced by some Riemannian metrics from the ambi- ent tangent bundle, was studied by authors like Abbassi, Boeckx, Cal- varuso, Kowalski, Munteanu, Park, Sekigawa, Sekizawa (see [1], [2], [4]-[7], [9], [12]-[14], [16]). In the most part of the papers, such as [4], [5] and [16], the metric considered on the tangent bundle TM was the Sasaki metric, but Boeckx noticed that the unit tangent bundle equipped with the induced Cheeger-Gromoll metric is isometric to the tangent sphere bun- dle T p1 M with the induced Sasaki metric. Then, in 2000, Kowalski and 2 Sekizawa showed how the geometry of the tangent sphere bundles depends on the radius (see [10]). -
WHAT IS a CONNECTION, and WHAT IS IT GOOD FOR? Contents 1. Introduction 2 2. the Search for a Good Directional Derivative 3 3. F
WHAT IS A CONNECTION, AND WHAT IS IT GOOD FOR? TIMOTHY E. GOLDBERG Abstract. In the study of differentiable manifolds, there are several different objects that go by the name of \connection". I will describe some of these objects, and show how they are related to each other. The motivation for many notions of a connection is the search for a sufficiently nice directional derivative, and this will be my starting point as well. The story will by necessity include many supporting characters from differential geometry, all of whom will receive a brief but hopefully sufficient introduction. I apologize for my ungrammatical title. Contents 1. Introduction 2 2. The search for a good directional derivative 3 3. Fiber bundles and Ehresmann connections 7 4. A quick word about curvature 10 5. Principal bundles and principal bundle connections 11 6. Associated bundles 14 7. Vector bundles and Koszul connections 15 8. The tangent bundle 18 References 19 Date: 26 March 2008. 1 1. Introduction In the study of differentiable manifolds, there are several different objects that go by the name of \connection", and this has been confusing me for some time now. One solution to this dilemma was to promise myself that I would some day present a talk about connections in the Olivetti Club at Cornell University. That day has come, and this document contains my notes for this talk. In the interests of brevity, I do not include too many technical details, and instead refer the reader to some lovely references. My main references were [2], [4], and [5]. -
Differential Topology
Differential Topology: Homework Set # 3. 1) Basic on tangent bundles. Recall that the tangent bundle TM to an extrinsicly defined manifold M k ⊂ Rn is defined to be: n n k n ∼ n TM := {(p, v) ∈ R × R | p ∈ M , v ∈ TpM ⊂ TpR = R } This exercise is designed to give you some practice working with tangent bundles. (i) Show that for a manifold M, the tangent bundle TM is well-defined, i.e. does not depend on the embedding M k ⊂ Rn. (ii) If M, N are a pair of smooth manifolds, show that T (M × N) is diffeomorphic to TM × TN. (iii) If f : M → R is a smooth, positive function, show that the corresponding scaling map φ : TM → TM defined by φ(p, v) = (p, f(p) · v) is a diffeomorphism. (iv) Show that the tangent bundle to S1 (the unit sphere) is diffeomorphic to the cylinder S1 × R. (v) Show that if Sn is an arbitrary sphere, then there exists a diffeomorphism between (TSn) × R and Sn × Rn+1. 2) Relatives of the tangent bundle. Closely associated to the tangent bundle are some other manifolds: (i) For an extrinsicly defined manifold M k ⊂ Rn, define the normal bundle as follows: n n k ⊥ n ∼ n N(M) := {(p, v) ∈ R × R | p ∈ M , v ∈ TpM ⊂ TpR = R } Show that N(M) is an n-dimensional manifold. (ii) Show that for the standard n-sphere Sn ⊂ Rn+1, the normal bundle N(Sn) is diffeomorphic to Sn × R. (iii) Consider the subset T 1M of TM consisting of pairs (p, v) ∈ TM with ||v|| = 1 (where || · || denotes the standard norm on Rn).