# Transparent connections

Size: px
Start display at page:

Download "Transparent connections"

Transcription

1

2 The abelian case A definition (M, g) is a closed Riemannian manifold, d = dim M. E M is a rank n complex vector bundle with a Hermitian metric (i.e. a U(n)-bundle). is a Hermitian (i.e. metric) connection on E. Definition: is said to be transparent if the parallel transport of along every closed geodesic of g is the identity. Problem: understand transparent connections. These connections are ghosts or invisible form the point of view of g. Obviously, transparent connections are invariant under gauge transformations.

3 The abelian case Motivation This is a typical inverse problem that relates to many others (DN maps, inverse spectral problems, derivatives of entropy...). When M is a bounded domain in R n with smooth boundary (or R n with suitable decay conditions), this problem has been studied by R. Novikov, Finch & Uhlmann, Sharafutdinov and G. Eskin among others.

4 The abelian case Here we will focus on the closed case. The role of the boundary will now be played by the dynamical properties of the geodesic flow of g. We can also replace the closed geodesics of g by another set of distinguished closed curves; for example the closed geodesics of an arbitrary affine connection on M. This finds applications in non-equilibrium statistical mechanics.

5 The abelian case We cannot expect to be able to understand transparent connections for an arbitrary g, but there are two important and opposite cases in which we can hope to be successful: g is a Zoll metric; g has negative curvature. In the closed case, there are results for M = S 2 and RP 2 with the round (Zoll) metric by R. Novikov and L. Mason (who introduced the term transparent connection ). appear naturally when trying to find finite dimensional families of solutions to the Yang-Mills-Higgs equations D A Φ = F A in (2 + 1)-space (Ward).

6 The abelian case Our setting g has negative curvature; E = M C n is a trivial vector bundle: this is not a serious restriction in the sense that our results will carry on to arbitrary bundles, but the assumption simplifies the presentation; d = 2, i.e. M is a surface. For E = M C n, the trivial connection is always transparent. One of the questions we wish to address is the following: Is the trivial connection the only transparent connection up to gauge equivalence?

7 The abelian case Connections A connection is given by a u(n)-valued 1-form A, which we view as a map A : TM u(n) which is linear in the velocities v T x M. Here = d + A. Recall that A and B are gauge equivalent if there exists a smooth function u : M U(n) such that B = u 1 Au + u 1 du. Thus A is gauge equivalent to the trivial connection iff there exists u : M U(n) such that du + Au = 0. Given γ : [a, b] M, the parallel transport along γ is obtained by solving the linear ODE: ṡ = A(γ, γ)s.

8 The abelian case The abelian case when n = 1, A = iθ, where θ Ω 1 (M). Transparent means θ 2πZ (T ) γ for every closed geodesic γ. A is gauge equivalent to the trivial connection iff there is u : M U(1) = S 1 such that 1 u du = iθ iff θ is closed and [θ/2π] H 1 (M, Z). Then, the question here is: does (T) = θ closed?

9 The abelian case The answer is YES. This follows (not immediately!) from the following result: if θ = 0 for all closed geodesics γ, then θ is exact. γ This was proved by Guillemin & Kazhdan (1980) for d = 2 and by Croke & Sharafutdinov (1998) for d 3.

10 Local uniqueness of the trivial connection A PDE classifying transparent connections Curvature Let F A be the curvature of = d + A. This is a u(n)-valued 2-form given by F A = da + A A. F A : M u(n), where is the Hodge star operator of (M, g). In other words, F A = ( F A )ω a, where ω a is the area form of (M, g). Let K be the Gaussian curvature of M. For each x M, ±i( F A ) K Id is a Hermitian matrix, which is positive definite for K < 0 and F A sufficiently small.

11 Local uniqueness of the trivial connection A PDE classifying transparent connections Local uniqueness Our first result is: Theorem A. Let (M, g) be a closed negatively curved orientable surface. Let A be a transparent connection on E = M C n, n 1. Then, if ±i( F A ) K Id is positive definite for all x M, A is gauge equivalent to the trivial connection. Question: Are there non-trivial transparent connections for n 2? Yes! Levi-Civita on T M TM, so the theorem is optimal. In fact there are many other non-trivial ghosts.

12 Local uniqueness of the trivial connection A PDE classifying transparent connections A classification result T transparent connections modulo gauge. SM is the unit circle bundle of M with canonical frame {X, H, V } where X is the geodesic vector field, V is the vertical vector field and H = [V, X ]. Let f : SM u(n) be a smooth function. Consider the PDE: H(f ) + VX (f ) = [X (f ), f ]. Let H 0 be the set of all solutions to the PDE for which there is u : SM U(n) such that f = u 1 V (u). Note that U(n) acts on H 0 by conjugation. Theorem B. There is a 1-1 correspondence between T and H 0 /U(n).

13 Local uniqueness of the trivial connection A PDE classifying transparent connections Degree one SU(2)-ghosts Suppose f depends only on the base point x M and takes values in su(2). Then the equation H(f ) + VX (f ) = [X (f ), f ] turns into 2 df = [df, f ]. For f non-constant, this implies f 2 = I and df = df f. Its solutions correspond precisely with holomorphic maps f : M CP 1. For any of them one can show that there is u : SM SU(2) such that f = u 1 V (u) (locally u = cos θi + sin θf and V = / θ). The Levi-Civita ghosts correspond to f being constant.

14 Obtaining new transparent connections Description of all the SU(2)-ghosts for SU(2) Suppose f H 0 and f = b 1 V (b) where b : SM SU(2), so f defines a transparent connection A. Let g : M su(2) be a smooth map with det g = 1 (i.e. g 2 = I ). We can always find a : SM SU(2) such that g = a 1 V (a). Let u := ab : SM SU(2) and let F := (ab) 1 V (ab). A key calculation shows that F H 0 iff - d A g = (d A g)g.

15 Obtaining new transparent connections Description of all the SU(2)-ghosts Description of all the SU(2)-ghosts The connection A induces an operator A and defines a holomorphic structure on the trivial bundle M C 2. - d A g = (d A g)g says precisely that the i-eigenspace of g is a A -holomorphic line bundle. Theorem C. Any transparent SU(2)-connection is obtained by applying a finite number of as described above.

16 A non-abelian cocycle The non-abelian Livsic theorem Cocycles Let SM be the unit sphere bundle of (M, g) and φ t : SM SM the geodesic flow. Let C : SM R U(n) be the unique solution to C t = A(φ t(x, v))c, C(x, v, 0) = Id. C is a U(n)-cocycle over φ t : C(x, v, t + s) = C(φ s (x, v), t)c(x, v, s). A transparent means that C(x, v, T ) = Id whenever φ T (x, v) = (x, v) (closed orbits).

17 A non-abelian cocycle The non-abelian Livsic theorem The non-abelian Livsic theorem A cocycle C : SM R U(n) is said to be cohomologically trivial if there exists a continuous u : SM U(n) such that C(x, v, t) = u(φ t (x, v))u 1 (x, v). The Livsic theorem (1972). A cocycle C is cohomologically trivial iff C(x, v, T ) = Id whenever φ T (x, v) = (x, v). Regularity (Nitica & Török, 1998). If C is C k, then u is C k ε for any small ε > 0 (for k = 1,, ω, we may take ε = 0).

18 A non-abelian cocycle The non-abelian Livsic theorem Summarizing: if A is transparent, there exists a smooth function u : SM U(n) such that C(x, v, t) = u(φ t (x, v))u 1 (x, v). Differentiating with respect to t and setting t = 0 we obtain: X (u) + Au = 0. Let V be the vertical vector field, i.e. the infinitesimal generator of the circle action of the bundle π : SM M. Goal: to show that V (u) = 0 for Theorem A, or that u must be a polynomial in the velocities for Theorem C (finite degree).

19 A non-abelian cocycle The non-abelian Livsic theorem Why V (u) = 0 implies Theorem A If V (u) = 0, then u is independent of v, so in fact u(x, v) = ũ(π(x, v)), for some smooth ũ : M U(n). But X (ũ π)(x, v) = dũ x (v) and X (u) + Au = 0 implies that A is gauge equivalent to the trivial connection! The heart of the argument is then to show that if u : SM U(n) satisfies the linear PDE X (u) + Au = 0, then V (u) = 0 under the hypothesis of Theorem A.

20 The operators of Guillemin and Kazhdan Modified operators Sketch of proof that V (u) = 0 Let M n (C) be the set of n n complex matrices. Given functions u, v : SM M n (C) we consider the inner product u, v = trace (u v ) dµ. SM The space L 2 (SM, M n (C)) decomposes orthogonally as a direct sum L 2 (SM, M n (C)) = n Z H n where iv acts as n Id on H n. We can write A = A 1 + A 1, where A 1 = (A iv (A))/2 H 1, A 1 = (A + iv (A))/2 H 1.

21 The operators of Guillemin and Kazhdan Modified operators Sketch of proof that V (u) = 0 The operators of Guillemin and Kazhdan Introduce the first order elliptic operators η + := (X ih)/2, η := (X + ih)/2. Clearly X = η + + η. We have η + : H n H n+1, η : H n H n 1, (η + ) = η.

22 The operators of Guillemin and Kazhdan Modified operators Sketch of proof that V (u) = 0 Modified operators To deal with X (u) + Au = 0, introduce the operators We also have µ + := η + + A 1,, µ := η + A 1. µ + : H n H n+1, µ : H n H n 1, (µ + ) = µ. X (u) + Au = 0 is now µ + (u) + µ (u) = 0. Calculating [µ +, µ ] one gets the key property: µ + (f ) 2 = µ (f ) 2 + (i F A Kn Id)f, f /2 µ (f ) 2 + c f 2, for f H n, n 1, c > 0 constant.

23 The operators of Guillemin and Kazhdan Modified operators Sketch of proof that V (u) = 0 V (u) = 0 Expand u = u n to get µ + (u n 1 ) + µ (u n+1 ) = 0. We have to show that u n = 0 for all n 0. The key property above shows that µ + (u n+1 ) 2 µ + (u n 1 ) 2 for n 2. But µ + (u n+1 ) 2 0 as n, thus µ + (u n 1 ) = 0 for all n 2. Since µ + is injective in H n for n 1, we have u n = 0 for all n 1 and we are done!

24 Higgs fields Suppose Φ : M u(n) is given. The pair (A, Φ) determines a cocycle C obtained by solving C t = {A(φ t(x, v)) + Φ(π φ t (x, v))}c, C(x, v, 0) = Id. (A, Φ) transparent means that C(x, v, T ) = Id whenever φ T (x, v) = (x, v) (closed orbits).

25 Transparent SU(2)-pairs can also be classified using a suitable Bäcklund transformation. There is a also a finiteness result for the Fourier series of the relevant u given by the Livsic theorem. However the Bäcklund transformation requires to work with SO(3) as structure group. To obtain transparent SU(2)-pairs of degree one needs to perform two Bäcklund SO(3)-transformations starting from the trivial pair.

26 Most of what was said above works for arbitrary bundles but we have the following topological constraint coming from the Livsic theorem suitably used: Theorem. Let E be a complex vector bundle over a closed negatively curved orientable surface of genus g. Then E admits a transparent connection if and only if 2 2g divides the degree of E (= the first Chern class of E).

### THE LONGITUDINAL KAM-COCYCLE OF A MAGNETIC FLOW

THE LONGITUDINAL KAM-COCYCLE OF A MAGNETIC FLOW GABRIEL P. PATERNAIN Abstract. Let M be a closed oriented surface of negative Gaussian curvature and let Ω be a non-exact 2-form. Let λ be a small positive

More information

### THE ATTENUATED RAY TRANSFORM FOR CONNECTIONS AND HIGGS FIELDS

THE ATTENUATED RAY TRANSFORM FOR CONNECTIONS AND HIGGS FIELDS GABRIEL P. PATERNAIN, MIKKO SALO, AND GUNTHER UHLMANN Abstract. We show that for a simple surface with boundary the attenuated ray transform

More information

### Inverse problems for connections

Inside Out II MSRI Publications Volume 60, 2012 Inverse problems for connections GABRIEL P. PATERNAIN We discuss various recent results related to the inverse problem of determining a unitary connection

More information

### LECTURE 26: THE CHERN-WEIL THEORY

LECTURE 26: THE CHERN-WEIL THEORY 1. Invariant Polynomials We start with some necessary backgrounds on invariant polynomials. Let V be a vector space. Recall that a k-tensor T k V is called symmetric if

More information

### SOME EXERCISES IN CHARACTERISTIC CLASSES

SOME EXERCISES IN CHARACTERISTIC CLASSES 1. GAUSSIAN CURVATURE AND GAUSS-BONNET THEOREM Let S R 3 be a smooth surface with Riemannian metric g induced from R 3. Its Levi-Civita connection can be defined

More information

### As always, the story begins with Riemann surfaces or just (real) surfaces. (As we have already noted, these are nearly the same thing).

An Interlude on Curvature and Hermitian Yang Mills As always, the story begins with Riemann surfaces or just (real) surfaces. (As we have already noted, these are nearly the same thing). Suppose we wanted

More information

### Metrics and Holonomy

Metrics and Holonomy Jonathan Herman The goal of this paper is to understand the following definitions of Kähler and Calabi-Yau manifolds: Definition. A Riemannian manifold is Kähler if and only if it

More information

### Recent progress on the explicit inversion of geodesic X-ray transforms

Recent progress on the explicit inversion of geodesic X-ray transforms François Monard Department of Mathematics, University of Washington. Geometric Analysis and PDE seminar University of Cambridge, May

More information

### INSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD

INSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD () Instanton (definition) (2) ADHM construction (3) Compactification. Instantons.. Notation. Throughout this talk, we will use the following notation:

More information

### EXISTENCE THEORY FOR HARMONIC METRICS

EXISTENCE THEORY FOR HARMONIC METRICS These are the notes of a talk given by the author in Asheville at the workshop Higgs bundles and Harmonic maps in January 2015. It aims to sketch the proof of the

More information

### Mathematical Research Letters 2, (1995) A VANISHING THEOREM FOR SEIBERG-WITTEN INVARIANTS. Shuguang Wang

Mathematical Research Letters 2, 305 310 (1995) A VANISHING THEOREM FOR SEIBERG-WITTEN INVARIANTS Shuguang Wang Abstract. It is shown that the quotients of Kähler surfaces under free anti-holomorphic involutions

More information

### Exercises in Geometry II University of Bonn, Summer semester 2015 Professor: Prof. Christian Blohmann Assistant: Saskia Voss Sheet 1

Assistant: Saskia Voss Sheet 1 1. Conformal change of Riemannian metrics [3 points] Let (M, g) be a Riemannian manifold. A conformal change is a nonnegative function λ : M (0, ). Such a function defines

More information

### Holomorphic line bundles

Chapter 2 Holomorphic line bundles In the absence of non-constant holomorphic functions X! C on a compact complex manifold, we turn to the next best thing, holomorphic sections of line bundles (i.e., rank

More information

### The X-ray transform for a non-abelian connection in two dimensions

The X-ray transform for a non-abelian connection in two dimensions David Finch Department of Mathematics Oregon State University Corvallis, OR, 97331, USA Gunther Uhlmann Department of Mathematics University

More information

### Pin (2)-monopole theory I

Intersection forms with local coefficients Osaka Medical College Dec 12, 2016 Pin (2) = U(1) j U(1) Sp(1) H Pin (2)-monopole equations are a twisted version of the Seiberg-Witten (U(1)-monopole) equations.

More information

### Topological Solitons from Geometry

Topological Solitons from Geometry Maciej Dunajski Department of Applied Mathematics and Theoretical Physics University of Cambridge Atiyah, Manton, Schroers. Geometric models of matter. arxiv:1111.2934.

More information

### 1. Geometry of the unit tangent bundle

1 1. Geometry of the unit tangent bundle The main reference for this section is [8]. In the following, we consider (M, g) an n-dimensional smooth manifold endowed with a Riemannian metric g. 1.1. Notations

More information

### LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI. 1. Maximal Tori

LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI 1. Maximal Tori By a torus we mean a compact connected abelian Lie group, so a torus is a Lie group that is isomorphic to T n = R n /Z n. Definition 1.1.

More information

### SYMPLECTIC MANIFOLDS, GEOMETRIC QUANTIZATION, AND UNITARY REPRESENTATIONS OF LIE GROUPS. 1. Introduction

SYMPLECTIC MANIFOLDS, GEOMETRIC QUANTIZATION, AND UNITARY REPRESENTATIONS OF LIE GROUPS CRAIG JACKSON 1. Introduction Generally speaking, geometric quantization is a scheme for associating Hilbert spaces

More information

### Donaldson and Seiberg-Witten theory and their relation to N = 2 SYM

Donaldson and Seiberg-Witten theory and their relation to N = SYM Brian Williams April 3, 013 We ve began to see what it means to twist a supersymmetric field theory. I will review Donaldson theory and

More information

### CHARACTERISTIC CLASSES

1 CHARACTERISTIC CLASSES Andrew Ranicki Index theory seminar 14th February, 2011 2 The Index Theorem identifies Introduction analytic index = topological index for a differential operator on a compact

More information

### MILNOR SEMINAR: DIFFERENTIAL FORMS AND CHERN CLASSES

MILNOR SEMINAR: DIFFERENTIAL FORMS AND CHERN CLASSES NILAY KUMAR In these lectures I want to introduce the Chern-Weil approach to characteristic classes on manifolds, and in particular, the Chern classes.

More information

### An Invitation to Geometric Quantization

An Invitation to Geometric Quantization Alex Fok Department of Mathematics, Cornell University April 2012 What is quantization? Quantization is a process of associating a classical mechanical system to

More information

### MATH 263: PROBLEM SET 1: BUNDLES, SHEAVES AND HODGE THEORY

MATH 263: PROBLEM SET 1: BUNDLES, SHEAVES AND HODGE THEORY 0.1. Vector Bundles and Connection 1-forms. Let E X be a complex vector bundle of rank r over a smooth manifold. Recall the following abstract

More information

### Morse theory and stable pairs

Richard A. SCGAS 2010 Joint with Introduction Georgios Daskalopoulos (Brown University) Jonathan Weitsman (Northeastern University) Graeme Wilkin (University of Colorado) Outline Introduction 1 Introduction

More information

### Ω Ω /ω. To these, one wants to add a fourth condition that arises from physics, what is known as the anomaly cancellation, namely that

String theory and balanced metrics One of the main motivations for considering balanced metrics, in addition to the considerations already mentioned, has to do with the theory of what are known as heterotic

More information

### Topic: First Chern classes of Kähler manifolds Mitchell Faulk Last updated: April 23, 2016

Topic: First Chern classes of Kähler manifolds itchell Faulk Last updated: April 23, 2016 We study the first Chern class of various Kähler manifolds. We only consider two sources of examples: Riemann surfaces

More information

### Poincaré Duality Angles on Riemannian Manifolds with Boundary

Poincaré Duality Angles on Riemannian Manifolds with Boundary Clayton Shonkwiler Department of Mathematics University of Pennsylvania June 5, 2009 Realizing cohomology groups as spaces of differential

More information

### Inversions of ray transforms on simple surfaces

Inversions of ray transforms on simple surfaces François Monard Department of Mathematics, University of Washington. June 09, 2015 Institut Henri Poincaré - Program on Inverse problems 1 / 42 Outline 1

More information

### Subgroups of Lie groups. Definition 0.7. A Lie subgroup of a Lie group G is a subgroup which is also a submanifold.

Recollections from finite group theory. The notion of a group acting on a set is extremely useful. Indeed, the whole of group theory arose through this route. As an example of the abstract power of this

More information

### The Theorem of Gauß-Bonnet in Complex Analysis 1

The Theorem of Gauß-Bonnet in Complex Analysis 1 Otto Forster Abstract. The theorem of Gauß-Bonnet is interpreted within the framework of Complex Analysis of one and several variables. Geodesic triangles

More information

### HYPERKÄHLER MANIFOLDS

HYPERKÄHLER MANIFOLDS PAVEL SAFRONOV, TALK AT 2011 TALBOT WORKSHOP 1.1. Basic definitions. 1. Hyperkähler manifolds Definition. A hyperkähler manifold is a C Riemannian manifold together with three covariantly

More information

### Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem

PETER B. GILKEY Department of Mathematics, University of Oregon Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem Second Edition CRC PRESS Boca Raton Ann Arbor London Tokyo Contents

More information

### Lecture 5. vector bundles, gauge theory

Lecture 5 vector bundles, gauge theory tangent bundle In Lecture 2, we defined the tangent space T p M at each point p on M. Let us consider a collection of tangent bundles over every point on M, TM =

More information

### SPECTRAL ASYMMETRY AND RIEMANNIAN GEOMETRY

SPECTRAL ASYMMETRY AND RIEMANNIAN GEOMETRY M. F. ATIYAH, V. K. PATODI AND I. M. SINGER 1 Main Theorems If A is a positive self-adjoint elliptic (linear) differential operator on a compact manifold then

More information

### Constructing compact 8-manifolds with holonomy Spin(7)

Constructing compact 8-manifolds with holonomy Spin(7) Dominic Joyce, Oxford University Simons Collaboration meeting, Imperial College, June 2017. Based on Invent. math. 123 (1996), 507 552; J. Diff. Geom.

More information

### LECTURE 10: THE ATIYAH-GUILLEMIN-STERNBERG CONVEXITY THEOREM

LECTURE 10: THE ATIYAH-GUILLEMIN-STERNBERG CONVEXITY THEOREM Contents 1. The Atiyah-Guillemin-Sternberg Convexity Theorem 1 2. Proof of the Atiyah-Guillemin-Sternberg Convexity theorem 3 3. Morse theory

More information

### Donaldson Invariants and Moduli of Yang-Mills Instantons

Donaldson Invariants and Moduli of Yang-Mills Instantons Lincoln College Oxford University (slides posted at users.ox.ac.uk/ linc4221) The ASD Equation in Low Dimensions, 17 November 2017 Moduli and Invariants

More information

### The spectral action for Dirac operators with torsion

The spectral action for Dirac operators with torsion Christoph A. Stephan joint work with Florian Hanisch & Frank Pfäffle Institut für athematik Universität Potsdam Tours, ai 2011 1 Torsion Geometry, Einstein-Cartan-Theory

More information

### This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail.

This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail. Author(s): Paternain, Gabriel P.; Salo, Mikko; Uhlmann, Gunther Title:

More information

### A little taste of symplectic geometry

A little taste of symplectic geometry Timothy Goldberg Thursday, October 4, 2007 Olivetti Club Talk Cornell University 1 2 What is symplectic geometry? Symplectic geometry is the study of the geometry

More information

### From holonomy reductions of Cartan geometries to geometric compactifications

From holonomy reductions of Cartan geometries to geometric compactifications 1 University of Vienna Faculty of Mathematics Berlin, November 11, 2016 1 supported by project P27072 N25 of the Austrian Science

More information

### CONSIDERATION OF COMPACT MINIMAL SURFACES IN 4-DIMENSIONAL FLAT TORI IN TERMS OF DEGENERATE GAUSS MAP

CONSIDERATION OF COMPACT MINIMAL SURFACES IN 4-DIMENSIONAL FLAT TORI IN TERMS OF DEGENERATE GAUSS MAP TOSHIHIRO SHODA Abstract. In this paper, we study a compact minimal surface in a 4-dimensional flat

More information

### Algebraic v.s. Analytic Point of View

Algebraic v.s. Analytic Point of View Ziwen Zhu September 19, 2015 In this talk, we will compare 3 different yet similar objects of interest in algebraic and complex geometry, namely algebraic variety,

More information

### Modern Geometric Structures and Fields

Modern Geometric Structures and Fields S. P. Novikov I.A.TaJmanov Translated by Dmitry Chibisov Graduate Studies in Mathematics Volume 71 American Mathematical Society Providence, Rhode Island Preface

More information

### Stratification of 3 3 Matrices

Stratification of 3 3 Matrices Meesue Yoo & Clay Shonkwiler March 2, 2006 1 Warmup with 2 2 Matrices { Best matrices of rank 2} = O(2) S 3 ( 2) { Best matrices of rank 1} S 3 (1) 1.1 Viewing O(2) S 3 (

More information

### arxiv: v2 [math.ap] 13 Sep 2015

THE X-RAY TRANSFORM FOR CONNECTIONS IN NEGATIVE CURVATURE COLIN GUILLARMOU, GABRIEL P. PATERNAIN, MIKKO SALO, AND GUNTHER UHLMANN arxiv:1502.04720v2 [math.ap] 13 Sep 2015 Abstract. We consider integral

More information

### Connections for noncommutative tori

Levi-Civita connections for noncommutative tori reference: SIGMA 9 (2013), 071 NCG Festival, TAMU, 2014 In honor of Henri, a long-time friend Connections One of the most basic notions in differential geometry

More information

### Quasi Riemann surfaces II. Questions, comments, speculations

Quasi Riemann surfaces II. Questions, comments, speculations Daniel Friedan New High Energy Theory Center, Rutgers University and Natural Science Institute, The University of Iceland dfriedan@gmail.com

More information

### A CONSTRUCTION OF TRANSVERSE SUBMANIFOLDS

UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA, FASCICULUS XLI 2003 A CONSTRUCTION OF TRANSVERSE SUBMANIFOLDS by J. Szenthe Abstract. In case of Riemannian manifolds isometric actions admitting submanifolds

More information

### NOTES ON NEWLANDER-NIRENBERG THEOREM XU WANG

NOTES ON NEWLANDER-NIRENBERG THEOREM XU WANG Abstract. In this short note we shall recall the classical Newler-Nirenberg theorem its vector bundle version. We shall also recall an L 2 -Hörmer-proof given

More information

### Qualifying Exams I, 2014 Spring

Qualifying Exams I, 2014 Spring 1. (Algebra) Let k = F q be a finite field with q elements. Count the number of monic irreducible polynomials of degree 12 over k. 2. (Algebraic Geometry) (a) Show that

More information

### CALCULUS ON MANIFOLDS. 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M =

CALCULUS ON MANIFOLDS 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M = a M T am, called the tangent bundle, is itself a smooth manifold, dim T M = 2n. Example 1.

More information

### LECTURE 16: CONJUGATE AND CUT POINTS

LECTURE 16: CONJUGATE AND CUT POINTS 1. Conjugate Points Let (M, g) be Riemannian and γ : [a, b] M a geodesic. Then by definition, exp p ((t a) γ(a)) = γ(t). We know that exp p is a diffeomorphism near

More information

### Symplectic critical surfaces in Kähler surfaces

Symplectic critical surfaces in Kähler surfaces Jiayu Li ( Joint work with X. Han) ICTP-UNESCO and AMSS-CAS November, 2008 Symplectic surfaces Let M be a compact Kähler surface, let ω be the Kähler form.

More information

### Math. Res. Lett. 13 (2006), no. 1, c International Press 2006 ENERGY IDENTITY FOR ANTI-SELF-DUAL INSTANTONS ON C Σ.

Math. Res. Lett. 3 (2006), no., 6 66 c International Press 2006 ENERY IDENTITY FOR ANTI-SELF-DUAL INSTANTONS ON C Σ Katrin Wehrheim Abstract. We establish an energy identity for anti-self-dual connections

More information

### Conformal field theory in the sense of Segal, modified for a supersymmetric context

Conformal field theory in the sense of Segal, modified for a supersymmetric context Paul S Green January 27, 2014 1 Introduction In these notes, we will review and propose some revisions to the definition

More information

### Lecture 4: Harmonic forms

Lecture 4: Harmonic forms Jonathan Evans 29th September 2010 Jonathan Evans () Lecture 4: Harmonic forms 29th September 2010 1 / 15 Jonathan Evans () Lecture 4: Harmonic forms 29th September 2010 2 / 15

More information

### THE GEOMETRY OF B-FIELDS. Nigel Hitchin (Oxford) Odense November 26th 2009

THE GEOMETRY OF B-FIELDS Nigel Hitchin (Oxford) Odense November 26th 2009 THE B-FIELD IN PHYSICS B = i,j B ij dx i dx j flux: db = H a closed three-form Born-Infeld action: det(g ij + B ij ) complexified

More information

### Vortex Equations on Riemannian Surfaces

Vortex Equations on Riemannian Surfaces Amanda Hood, Khoa Nguyen, Joseph Shao Advisor: Chris Woodward Vortex Equations on Riemannian Surfaces p.1/36 Introduction Yang Mills equations: originated from electromagnetism

More information

### Eigenvalues and Eigenvectors

/88 Chia-Ping Chen Department of Computer Science and Engineering National Sun Yat-sen University Linear Algebra Eigenvalue Problem /88 Eigenvalue Equation By definition, the eigenvalue equation for matrix

More information

### Summary of Prof. Yau s lecture, Monday, April 2 [with additional references and remarks] (for people who missed the lecture)

Building Geometric Structures: Summary of Prof. Yau s lecture, Monday, April 2 [with additional references and remarks] (for people who missed the lecture) A geometric structure on a manifold is a cover

More information

### Math 230a Final Exam Harvard University, Fall Instructor: Hiro Lee Tanaka

Math 230a Final Exam Harvard University, Fall 2014 Instructor: Hiro Lee Tanaka 0. Read me carefully. 0.1. Due Date. Per university policy, the official due date of this exam is Sunday, December 14th, 11:59

More information

### 4.2. ORTHOGONALITY 161

4.2. ORTHOGONALITY 161 Definition 4.2.9 An affine space (E, E ) is a Euclidean affine space iff its underlying vector space E is a Euclidean vector space. Given any two points a, b E, we define the distance

More information

### The Strominger Yau Zaslow conjecture

The Strominger Yau Zaslow conjecture Paul Hacking 10/16/09 1 Background 1.1 Kähler metrics Let X be a complex manifold of dimension n, and M the underlying smooth manifold with (integrable) almost complex

More information

### GEOMETRIC QUANTIZATION

GEOMETRIC QUANTIZATION 1. The basic idea The setting of the Hamiltonian version of classical (Newtonian) mechanics is the phase space (position and momentum), which is a symplectic manifold. The typical

More information

### 1. Plurisubharmonic functions and currents The first part of the homework studies some properties of PSH functions.

MATH 263: PROBLEM SET 2: PSH FUNCTIONS, HORMANDER S ESTIMATES AND VANISHING THEOREMS 1. Plurisubharmonic functions and currents The first part of the homework studies some properties of PSH functions.

More information

### Hyperkähler geometry lecture 3

Hyperkähler geometry lecture 3 Misha Verbitsky Cohomology in Mathematics and Physics Euler Institute, September 25, 2013, St. Petersburg 1 Broom Bridge Here as he walked by on the 16th of October 1843

More information

### RICCI SOLITONS ON COMPACT KAHLER SURFACES. Thomas Ivey

RICCI SOLITONS ON COMPACT KAHLER SURFACES Thomas Ivey Abstract. We classify the Kähler metrics on compact manifolds of complex dimension two that are solitons for the constant-volume Ricci flow, assuming

More information

### k=0 /D : S + S /D = K 1 2 (3.5) consistently with the relation (1.75) and the Riemann-Roch-Hirzebruch-Atiyah-Singer index formula

20 VASILY PESTUN 3. Lecture: Grothendieck-Riemann-Roch-Hirzebruch-Atiyah-Singer Index theorems 3.. Index for a holomorphic vector bundle. For a holomorphic vector bundle E over a complex variety of dim

More information

### Eigenvalues and eigenfunctions of the Laplacian. Andrew Hassell

Eigenvalues and eigenfunctions of the Laplacian Andrew Hassell 1 2 The setting In this talk I will consider the Laplace operator,, on various geometric spaces M. Here, M will be either a bounded Euclidean

More information

### L 2 extension theorem for sections defined on non reduced analytic subvarieties

L 2 extension theorem for sections defined on non reduced analytic subvarieties Jean-Pierre Demailly Institut Fourier, Université de Grenoble Alpes & Académie des Sciences de Paris Conference on Geometry

More information

### Theorem 2. Let n 0 3 be a given integer. is rigid in the sense of Guillemin, so are all the spaces ḠR n,n, with n n 0.

This monograph is motivated by a fundamental rigidity problem in Riemannian geometry: determine whether the metric of a given Riemannian symmetric space of compact type can be characterized by means of

More information

### AN INTRODUCTION TO THE MASLOV INDEX IN SYMPLECTIC TOPOLOGY

1 AN INTRODUCTION TO THE MASLOV INDEX IN SYMPLECTIC TOPOLOGY Andrew Ranicki and Daniele Sepe (Edinburgh) http://www.maths.ed.ac.uk/ aar Maslov index seminar, 9 November 2009 The 1-dimensional Lagrangians

More information

### Vanishing theorems and holomorphic forms

Vanishing theorems and holomorphic forms Mihnea Popa Northwestern AMS Meeting, Lansing March 14, 2015 Holomorphic one-forms and geometry X compact complex manifold, dim C X = n. Holomorphic one-forms and

More information

### LECTURE 2: SYMPLECTIC VECTOR BUNDLES

LECTURE 2: SYMPLECTIC VECTOR BUNDLES WEIMIN CHEN, UMASS, SPRING 07 1. Symplectic Vector Spaces Definition 1.1. A symplectic vector space is a pair (V, ω) where V is a finite dimensional vector space (over

More information

### The Spectral Geometry of the Standard Model

Ma148b Spring 2016 Topics in Mathematical Physics References A.H. Chamseddine, A. Connes, M. Marcolli, Gravity and the Standard Model with Neutrino Mixing, Adv. Theor. Math. Phys., Vol.11 (2007) 991 1090

More information

### HARMONIC MAPS INTO GRASSMANNIANS AND A GENERALIZATION OF DO CARMO-WALLACH THEOREM

Proceedings of the 16th OCU International Academic Symposium 2008 OCAMI Studies Volume 3 2009, pp.41 52 HARMONIC MAPS INTO GRASSMANNIANS AND A GENERALIZATION OF DO CARMO-WALLACH THEOREM YASUYUKI NAGATOMO

More information

### A geometric solution of the Kervaire Invariant One problem

A geometric solution of the Kervaire Invariant One problem Petr M. Akhmet ev 19 May 2009 Let f : M n 1 R n, n = 4k + 2, n 2 be a smooth generic immersion of a closed manifold of codimension 1. Let g :

More information

### arxiv: v1 [math.sg] 6 Nov 2015

A CHIANG-TYPE LAGRANGIAN IN CP ANA CANNAS DA SILVA Abstract. We analyse a simple Chiang-type lagrangian in CP which is topologically an RP but exhibits a distinguishing behaviour under reduction by one

More information

### Infinitesimal Einstein Deformations. Kähler Manifolds

on Nearly Kähler Manifolds (joint work with P.-A. Nagy and U. Semmelmann) Gemeinsame Jahrestagung DMV GDM Berlin, March 30, 2007 Nearly Kähler manifolds Definition and first properties Examples of NK manifolds

More information

### Differential Topology Final Exam With Solutions

Differential Topology Final Exam With Solutions Instructor: W. D. Gillam Date: Friday, May 20, 2016, 13:00 (1) Let X be a subset of R n, Y a subset of R m. Give the definitions of... (a) smooth function

More information

### Background on c-projective geometry

Second Kioloa Workshop on C-projective Geometry p. 1/26 Background on c-projective geometry Michael Eastwood [ following the work of others ] Australian National University Second Kioloa Workshop on C-projective

More information

### Complex manifolds, Kahler metrics, differential and harmonic forms

Complex manifolds, Kahler metrics, differential and harmonic forms Cattani June 16, 2010 1 Lecture 1 Definition 1.1 (Complex Manifold). A complex manifold is a manifold with coordinates holomorphic on

More information

### Elementary realization of BRST symmetry and gauge fixing

Elementary realization of BRST symmetry and gauge fixing Martin Rocek, notes by Marcelo Disconzi Abstract This are notes from a talk given at Stony Brook University by Professor PhD Martin Rocek. I tried

More information

### The existence of light-like homogeneous geodesics in homogeneous Lorentzian manifolds. Sao Paulo, 2013

The existence of light-like homogeneous geodesics in homogeneous Lorentzian manifolds Zdeněk Dušek Sao Paulo, 2013 Motivation In a previous project, it was proved that any homogeneous affine manifold (and

More information

### Some brief notes on the Kodaira Vanishing Theorem

Some brief notes on the Kodaira Vanishing Theorem 1 Divisors and Line Bundles, according to Scott Nollet This is a huge topic, because there is a difference between looking at an abstract variety and local

More information

### N = 2 supersymmetric gauge theory and Mock theta functions

N = 2 supersymmetric gauge theory and Mock theta functions Andreas Malmendier GTP Seminar (joint work with Ken Ono) November 7, 2008 q-series in differential topology Theorem (M-Ono) The following q-series

More information

### η = (e 1 (e 2 φ)) # = e 3

Research Statement My research interests lie in differential geometry and geometric analysis. My work has concentrated according to two themes. The first is the study of submanifolds of spaces with riemannian

More information

### Riemannian Curvature Functionals: Lecture III

Riemannian Curvature Functionals: Lecture III Jeff Viaclovsky Park City Mathematics Institute July 18, 2013 Lecture Outline Today we will discuss the following: Complete the local description of the moduli

More information

### Geometry and the Kato square root problem

Geometry and the Kato square root problem Lashi Bandara Centre for Mathematics and its Applications Australian National University 7 June 2013 Geometric Analysis Seminar University of Wollongong Lashi

More information

### Holonomy groups. Thomas Leistner. School of Mathematical Sciences Colloquium University of Adelaide, May 7, /15

Holonomy groups Thomas Leistner School of Mathematical Sciences Colloquium University of Adelaide, May 7, 2010 1/15 The notion of holonomy groups is based on Parallel translation Let γ : [0, 1] R 2 be

More information

### Basic Concepts of Group Theory

Chapter 1 Basic Concepts of Group Theory The theory of groups and vector spaces has many important applications in a number of branches of modern theoretical physics. These include the formal theory of

More information

### SOME ASPECTS ON CIRCLES AND HELICES IN A COMPLEX PROJECTIVE SPACE. Toshiaki Adachi* and Sadahiro Maeda

Mem. Fac. Sci. Eng. Shimane Univ. Series B: Mathematical Science 32 (1999), pp. 1 8 SOME ASPECTS ON CIRCLES AND HELICES IN A COMPLEX PROJECTIVE SPACE Toshiaki Adachi* and Sadahiro Maeda (Received December

More information

### Projective space and twistor theory

Hayama Symposium on Complex Analysis in Several Variables XVII p. 1/19 Projective space and twistor theory Michael Eastwood [ Toby Bailey Robin Graham Paul Baird Hubert Goldschmidt ] Australian National

More information

### UNIVERSAL UNFOLDINGS OF LAURENT POLYNOMIALS AND TT STRUCTURES. Claude Sabbah

UNIVERSAL UNFOLDINGS OF LAURENT POLYNOMIALS AND TT STRUCTURES AUGSBURG, MAY 2007 Claude Sabbah Introduction Let f : (C ) n C be a Laurent polynomial, that I assume to be convenient and non-degenerate,

More information

### TWISTOR AND KILLING FORMS IN RIEMANNIAN GEOMETRY

TWISTOR AND KILLING FORMS IN RIEMANNIAN GEOMETRY Andrei Moroianu CNRS - Ecole Polytechnique Palaiseau Prague, September 1 st, 2004 joint work with Uwe Semmelmann Plan of the talk Algebraic preliminaries

More information

### DIFFERENTIAL FORMS AND COHOMOLOGY

DIFFERENIAL FORMS AND COHOMOLOGY ONY PERKINS Goals 1. Differential forms We want to be able to integrate (holomorphic functions) on manifolds. Obtain a version of Stokes heorem - a generalization of the

More information

### Invariant Nonholonomic Riemannian Structures on Three-Dimensional Lie Groups

Invariant Nonholonomic Riemannian Structures on Three-Dimensional Lie Groups Dennis I. Barrett Geometry, Graphs and Control (GGC) Research Group Department of Mathematics, Rhodes University Grahamstown,

More information

### QUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday August 31, 2010 (Day 1)

QUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday August 31, 21 (Day 1) 1. (CA) Evaluate sin 2 x x 2 dx Solution. Let C be the curve on the complex plane from to +, which is along

More information