DOCSLIB.ORG

Spectral Geometry

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Spectral Geometry Notes for Spectral Geometry by Yaiza Canzani. 2016 CRM Summer School in Quebec City. Laval University. 3 Abstract I wrote these lectures using the material in set of notes \Analysis on manifolds via the Laplacian" available at http://www.math.harvard.edu/ canzani/docs/Laplacian.pdf. The material there corresponds to a year long course, so it contains more topics and many more details. The following references were important sources for these notes: Eigenvalues in Riemannian geometry. By I. Chavel. • Old and new aspects in Spectral Geometry. By M. Craiveanu, M. Puta and T. Ras- • sias. The Laplacian on a Riemannian manifold. By S. Rosenberg. • Local and global analysis of eigenfunctions on Riemannian manifolds. By S. Zelditch. • Enjoy!! Syllabus Abstract . 3 1 What makes the Laplacian special? 7 1.1 Daily life problems . 7 1.2 You want to solve the Helmholtz equation . 9 1.3 Inverse problem: Can you hear the shape of a drum? . 10 1.3.1 Hearing the length of a guitar string . 12 1.4 Direct problems . 14 1.4.1 The first eigenvalue: Rayleigh Conjecture . 14 1.4.2 Counting eigenvalues: Weyl's Law . 15 1.5 A hard problem: understanding the eigenfunctions . 18 2 The Laplacian on a Riemannian manifold 21 2.1 Definition of the Laplacian on a manifold . 21 2.1.1 Definition of a manifold . 21 2.1.2 Definition of a Riemannian metric . 24 2.1.3 Definition of the Laplacian . 27 2.2 Spectral Theory for the Laplacian . 28 2.2.1 Spectral Theorem . 30 2.2.2 Eigenvalue characterization (Exercise 4) . 30 2.3 Examples of eigenfunctions and eigenvalues . 31 2.3.1 Circle . 31 2.3.2 Torus . 31 2.3.3 Eigenfunctions on the sphere (Exercise 3) . 32 3 Hearing the geometry of a manifold 35 3.1 Heat Equation . 35 3.1.1 Solutions to Heat equation (Exercise 4) . 35 3.1.2 Definition of the fundamental solution . 35 3.2 Weyl's Law and other high energy asymptotics . 38 3.3 Isospectral manifolds . 40 4 Exercises 43 4.1 Exercise 0: Recovering the eigenvalues of a guitar string . 43 4.2 Exercise 1: Weyl's Law for a rectangle . 43 4.3 Exercise 2: Weyl's law on the Torus . 44 4.4 Exericise 3: Eigenfunctions on the Sphere . 44 4.5 Exercise 4: Characterization of eigenvalues . 45 4.6 Exercise 5: Temperature decreases with time and solutions are unique . 46 4.7 Exercise 5: Nodal domains and first eigenfunctions . 46 CHAPTER 1 What makes the Laplacian special? In this chapter we motivate the study of the Laplace operator. To simplify exposition, we do this by concentrating on planar domains. 1.1 Daily life problems n 1 Let Ω R be a connected domain and consider the operator ∆ acting on C (Ω) that ⊂ simply differentiates a function ' C1(Ω) two times with respect to each position 2 variable: n X @2' ∆' = : @x2 i=1 i Figure: Pierre-Simon de Laplace 8 What makes the Laplacian special? Here are some examples where the Laplacian plays a key role: Heat diffusion. If you are interested in understanding how would heat propagate n along Ω R then you should solve the Heat Equation ⊂ 1 @ ∆u(x; t) = u(x; t) c @t where c is the conductivity of the material of which Ω is made of, and u(x; t) is the temperature at the point x Ω at time t. 2 You could also think you have an insulated region Ω (it could be a wire, a ball, etc.) and apply certain given temperatures on the edge @Ω. If you want to know what the temperature will be after a long enough period of time (that is, the steady state temperature distribution), then you need to find a solution of the heat equation that be independent of time. The steady state temperature solution will be a function u(x1; : : : ; xn; t) such that ∆u = 0: Wave propagation. Now, instead of applying heat to the surface suppose you cover it with a thin layer of some fluid and you wish to describe the motion of the surface of the fluid. Then you will need to solve the Wave equation 1 @2 ∆u(x; t) = u(x; t) c @t2 where pc is the speed of sound in your fluid, and u(x; t) denotes the height of the wave above the point x at time t. You could also think of your domain Ω as the membrane of a drum, in which case its boundary @Ω would be attached to the rim of the drum. Suppose you want to study what will happen with the vibration you would generate if you hit it. Then, you @2 should also solve the wave equation ∆u(x; t) = @2t u(x; t) for your drum, but this time you want to make sure that you take into account that the border of the membrane is fixed. Thus, you should also ask your solution to satisfy u(x; t) = 0 for all points x @Ω. 2 Quantum particles. If you are a bit more eccentric and wish to see how a quantum particle moves inside Ω (under the assumption that there are no external forces) then you need to solve the Schr¨odingerEquation 2 ~ @ ∆u(x; t) = i~ u(x; t) −2m @t where ~ is Planck's constant and m is the mass of the free particle. Normalizing u so that u( ; t) 2 = 1 one interprets u(x; t) as a probability density. That is, if k · kL (Ω) A Ω then the probability that your quantum particle be inside A at time t is given ⊂ by R u(x; t) 2dx: A j j n Why not another operator? The Laplacian on R commutes with translations and rotations. That is, if T is a translation or rotation then ∆(' T ) = (∆') T . Something ◦ ◦ 1.2 You want to solve the Helmholtz equation 9 more striking occurs, if S is any operator that commutes with translations and rotations Pm j then there exist coefficients a1; : : : ; am making S = j=1 aj ∆ . Therefore, it is not surprising that the Laplacian will be a main star in any process whose underlying physics are independent of position and direction such as heat diffusion and wave propagation n in R . 1.2 You want to solve the Helmholtz equation There are of course many more problems involving the Laplacian, but we will focus on these ones to stress the importance of solving the eigenvalue problem (also known as Helmholtz equation) ∆' = λϕ. − It is clear that if one wants to study harmonic functions then one needs to solve the equation ∆' = λϕ with λ = 0: − So the need for understanding solutions of the Helmholtz equation for problems such as the static electric field or the steady-state fluid flow is straightforward. In order to attack the heat diffusion, wave propagation and Schr¨odingerproblems described above, a standard method (inspired by Stone-Weierstrass Theorem) is to look for solutions u(x; t) of the form u(x; t) = α(t)'(x). For instance if you do this and look at the Heat equation then you must have ∆'(x) α0(t) = x Ω; t > 0: '(x) α(t) 2 This shows that there must exist a λ R such that 2 α0 = λα and ∆' = λϕ. − − Therefore ' must be an eigenfunction of the Laplacian with eigenvalue λ and α(t) = −λt −λ t e . Once you have these particular solutions uk = e k 'k you use the superposition principle to write a general solution X −λkt u(x; t) = ake 'k(x) k where the coefficients ak are chosen depending on the initial conditions. You could do the same with the wave equation (we do it in detail for a guitar string in Section 1.3.1) or with the Schr¨odingerequation and you will also find particular solutions of the form uk(x; t) = αk(t)'k(x) with 8 e−λkt Heat eqn; <> p i λ t ∆'k = λk'k and αk(t) = e k Wave eqn; − > :eiλkt Schr¨odingereqn: 10 What makes the Laplacian special? 1.3 Inverse problem: Can you hear the shape of a drum? Inverse problem: If I know (more or less) the Laplace eigenvalues of a domain, what can I deduce of its geometry? Suppose you have perfect pitch. Could you derive the shape of a drum from the music you hear from it? More generally, can you determine the structural soundness of an object by listening to its vibrations? This question was first posed by Schuster in 1882. As Berger says in his book A panoramic view of Riemannian Geometry, \Already in the middle ages bell makers knew how to detect invisible cracks by sounding a bell on the ground before lifting it up to the belfry. How can one test the resistance to vibrations of large modern structures by non- destructive essays?... A small crack will not only change the boundary shape of our domain, one side of the crack will strike the other during vibrations invalidating our use of the simple linear wave equation. On the other hand, heat will presumably not leak out of a thin crack very quickly, so perhaps the heat equation will still provide a reasonable approximation for a short time..." An infinite sequence of numbers determines via Fourier analysis an integrable function. It wouldn't be that crazy if an infinite sequences of eigenvalues would determine the shape of the domain. Unfortunately, the answer to the question can you hear the shape of a drum? is no.
Load more

Recommended publications
  • Spectral Geometry Spring 2016
    Spectral Geometry Spring 2016 Michael Magee Lecture 5: The resolvent. Last time we proved that the Laplacian has a closure whose domain is the Sobolev space H2(M)= u L2(M): u L2(M), ∆u L2(M) . { 2 kr k2 2 } We view the Laplacian as an unbounded densely defined self-adjoint operator on L2 with domain H2. We also stated the spectral theorem. We’ll now ask what is the nature of the pro- jection valued measure associated to the Laplacian (its ‘spectral decomposition’). Until otherwise specified, suppose M is a complete Riemannian manifold. Definition 1. The resolvent set of the Laplacian is the set of λ C such that λI ∆isa bijection of H2 onto L2 with a boundedR inverse (with respect to L2 norms)2 − 1 2 2 R(λ) (λI ∆)− : L (M) H (M). ⌘ − ! The operator R(λ) is called the resolvent of the Laplacian at λ. We write spec(∆) = C −R and call it the spectrum. Note that the projection valued measure of ∆is supported in R 0. Otherwise one can find a function in H2(M)where ∆ , is negative, but since can be≥ approximated in Sobolev norm by smooth functions, thish will contradicti ∆ , = g( , )Vol 0. (1) h i ˆ r r ≥ Note then the spectral theorem implies that the resolvent set contains C R 0 (use the Borel functional calculus). − ≥ Theorem 2 (Stone’s formula [1, VII.13]). Let P be the projection valued measure on R associated to the Laplacan by the spectral theorem. The following formula relates the resolvent to P b 1 1 lim(2⇡i)− [R(λ + i✏) R(λ i✏)] dλ = P[a,b] + P(a,b) .
    [Show full text]
  • Isospectralization, Or How to Hear Shape, Style, and Correspondence
    Isospectralization, or how to hear shape, style, and correspondence Luca Cosmo Mikhail Panine Arianna Rampini University of Venice Ecole´ Polytechnique Sapienza University of Rome luca.cosmo@unive.it mpanine@lix.polytechnique.fr rampini@di.uniroma1.it Maks Ovsjanikov Michael M. Bronstein Emanuele Rodola` Ecole´ Polytechnique Imperial College London / USI Sapienza University of Rome maks@lix.polytechnique.fr m.bronstein@imperial.ac.uk emanuele.rodola@uniroma1.it Initialization Reconstruction Abstract opt. target The question whether one can recover the shape of a init. geometric object from its Laplacian spectrum (‘hear the shape of the drum’) is a classical problem in spectral ge- ometry with a broad range of implications and applica- tions. While theoretically the answer to this question is neg- ative (there exist examples of iso-spectral but non-isometric manifolds), little is known about the practical possibility of using the spectrum for shape reconstruction and opti- mization. In this paper, we introduce a numerical proce- dure called isospectralization, consisting of deforming one shape to make its Laplacian spectrum match that of an- other. We implement the isospectralization procedure using modern differentiable programming techniques and exem- plify its applications in some of the classical and notori- Without isospectralization With isospectralization ously hard problems in geometry processing, computer vi- sion, and graphics such as shape reconstruction, pose and Figure 1. Top row: Mickey-from-spectrum: we recover the shape of Mickey Mouse from its first 20 Laplacian eigenvalues (shown style transfer, and dense deformable correspondence. in red in the leftmost plot) by deforming an initial ellipsoid shape; the ground-truth target embedding is shown as a red outline on top of our reconstruction.
    [Show full text]
  • Arxiv:1908.09677V4 [Math.AG] 13 Jul 2021 6 Hr Ro Fterm12.1 Theorem of Proof Third a 16
    AN ANALYTIC VERSION OF THE LANGLANDS CORRESPONDENCE FOR COMPLEX CURVES PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN In memory of Boris Dubrovin Abstract. The Langlands correspondence for complex curves is traditionally for- mulated in terms of sheaves rather than functions. Recently, Langlands asked whether it is possible to construct a function-theoretic version. In this paper we use the algebra of commuting global differential operators (quantum Hitchin Hamilto- nians and their complex conjugates) on the moduli space of G-bundles of a complex algebraic curve to formulate a function-theoretic correspondence. We conjecture the existence of a canonical self-adjoint extension of the symmetric part of this al- gebra acting on an appropriate Hilbert space and link its spectrum with the set of opers for the Langlands dual group of G satisfying a certain reality condition, as predicted earlier by Teschner. We prove this conjecture for G = GL1 and in the simplest non-abelian case. Contents 1. Introduction 2 Part I 8 2. Differential operators on line bundles 8 3. Differential operators on BunG 11 4. The spectrum and opers 15 5. The abelian case 19 6. Bundles with parabolic structures 26 1 7. The case of SL2 and P with marked points 28 arXiv:1908.09677v4 [math.AG] 13 Jul 2021 8. Proofs of two results 30 Part II 33 9. Darboux operators 34 10. EigenfunctionsandmonodromyforDarbouxoperators 38 11. Essentially self-adjoint algebras of unbounded operators 43 12. The main theorem 49 13. Generalized Sobolev and Schwartz spaces attached to the operator L. 49 14. Proof of Theorem 12.1 60 15.
    [Show full text]
  • Singular Value Decomposition (SVD)
    San José State University Math 253: Mathematical Methods for Data Visualization Lecture 5: Singular Value Decomposition (SVD) Dr. Guangliang Chen Outline • Matrix SVD Singular Value Decomposition (SVD) Introduction We have seen that symmetric matrices are always (orthogonally) diagonalizable. That is, for any symmetric matrix A ∈ Rn×n, there exist an orthogonal matrix Q = [q1 ... qn] and a diagonal matrix Λ = diag(λ1, . , λn), both real and square, such that A = QΛQT . We have pointed out that λi’s are the eigenvalues of A and qi’s the corresponding eigenvectors (which are orthogonal to each other and have unit norm). Thus, such a factorization is called the eigendecomposition of A, also called the spectral decomposition of A. What about general rectangular matrices? Dr. Guangliang Chen | Mathematics & Statistics, San José State University3/22 Singular Value Decomposition (SVD) Existence of the SVD for general matrices Theorem: For any matrix X ∈ Rn×d, there exist two orthogonal matrices U ∈ Rn×n, V ∈ Rd×d and a nonnegative, “diagonal” matrix Σ ∈ Rn×d (of the same size as X) such that T Xn×d = Un×nΣn×dVd×d. Remark. This is called the Singular Value Decomposition (SVD) of X: • The diagonals of Σ are called the singular values of X (often sorted in decreasing order). • The columns of U are called the left singular vectors of X. • The columns of V are called the right singular vectors of X. Dr. Guangliang Chen | Mathematics & Statistics, San José State University4/22 Singular Value Decomposition (SVD) * * b * b (n>d) b b b * b = * * = b b b * (n<d) * b * * b b Dr.
    [Show full text]
  • Arxiv:2107.01242V1 [Math.OA] 2 Jul 2021 Euneo Ievle ( Eigenvalues of Sequence a Eetdacrigt T Agbac Utpiiy Y[ by Multiplicity
    CONNES’ INTEGRATION AND WEYL’S LAWS RAPHAEL¨ PONGE Abstract. This paper deal with some questions regarding the notion of integral in the frame- work of Connes’s noncommutative geometry. First, we present a purely spectral theoretic con- struction of Connes’ integral. This answers a question of Alain Connes. We also deal with the compatibility of Dixmier traces with Lebesgue’s integral. This answers another question of Alain Connes. We further clarify the relationship of Connes’ integration with Weyl’s laws for compact operators and Birman-Solomyak’s perturbation theory. We also give a ”soft proof” of Birman-Solomyak’s Weyl’s law for negative order pseudodifferential operators on closed mani- fold. This Weyl’s law yields a stronger form of Connes’ trace theorem. Finally, we explain the relationship between Connes’ integral and semiclassical Weyl’s law for Schr¨odinger operators. This is an easy consequence of the Birman-Schwinger principle. We thus get a neat link between noncommutative geometry and semiclassical analysis. 1. Introduction The quantized calculus of Connes [18] aims at translating the main tools of the classical infin- itesimal calculus into the operator theoretic language of quantum mechanics. As an Ansatz the integral in this setup should be a positive trace on the weak trace class L1,∞ (see Section 2). Natural choices are given by the traces Trω of Dixmier [23] (see also [18, 34] and Section 2). These traces are associated with extended limits. Following Connes [18] we say that an operator A L is measurable when the value of Tr (A) is independent of the extended limit.
    [Show full text]
  • Heat Kernels and Function Theory on Metric Measure Spaces
    Heat kernels and function theory on metric measure spaces Alexander Grigor’yan Imperial College London SW7 2AZ United Kingdom and Institute of Control Sciences of RAS Moscow, Russia email: a.grigoryan@ic.ac.uk Contents 1 Introduction 2 2 Definition of a heat kernel and examples 5 3 Volume of balls 8 4 Energy form 10 5 Besov spaces and energy domain 13 5.1 Besov spaces in Rn ............................ 13 5.2 Besov spaces in a metric measure space . 14 5.3 Identification of energy domains and Besov spaces . 15 5.4 Subordinated heat kernel ......................... 19 6 Intrinsic characterization of walk dimension 22 7 Inequalities for walk dimension 23 8 Embedding of Besov spaces into H¨olderspaces 29 9 Bessel potential spaces 30 1 1 Introduction The classical heat kernel in Rn is the fundamental solution to the heat equation, which is given by the following formula 1 x y 2 pt (x, y) = exp | − | . (1.1) n/2 t (4πt) − 4 ! x y 2 It is worth observing that the Gaussian term exp | − | does not depend in n, − 4t n/2 whereas the other term (4πt)− reflects the dependence of the heat kernel on the underlying space via its dimension. The notion of heat kernel extends to any Riemannian manifold M. In this case, the heat kernel pt (x, y) is the minimal positive fundamental solution to the heat ∂u equation ∂t = ∆u where ∆ is the Laplace-Beltrami operator on M, and it always exists (see [11], [13], [16]). Under certain assumptions about M, the heat kernel can be estimated similarly to (1.1).
    [Show full text]
  • WEYL's LAW on RIEMANNIAN MANIFOLDS Contents 1. the Role of Weyl's Law in the Ultraviolet Catastrophe 1 2. Riemannian Manifol
    WEYL'S LAW ON RIEMANNIAN MANIFOLDS SETH MUSSER Abstract. We motivate Weyl's law by demonstrating the relevance of the distribution of eigenvalues of the Laplacian to the ultraviolet catastrophe of physics. We then introduce Riemannian geometry, define the Laplacian on a general Riemannian manifold, and find geometric analogs of various concepts in Rn, along the way using S2 as a clarifying example. Finally, we use analogy with Rn and the results we have built up to prove Weyl's law, making sure at each step to demonstrate the physical significance of any major ideas. Contents 1. The Role of Weyl's Law in the Ultraviolet Catastrophe 1 2. Riemannian Manifolds 3 3. Weyl's Law 11 4. Acknowledgements 20 References 20 1. The Role of Weyl's Law in the Ultraviolet Catastrophe In the year 1900 Lord Rayleigh used the equipartition theorem of thermodynamics to de- duce the famous Rayleigh-Jeans law of radiation. Rayleigh began with an idealized physical concept of the blackbody; a body which is a perfect absorber of electromagnetic radiation and which radiates all the energy it absorbs independent of spatial direction. We sketch his proof to find the amount of radiation energy emitted by the blackbody at a given frequency. Take a blackbody in the shape of a cube for definiteness, and assume it is made of con- ducting material and is at a temperature T . We will denote the cube by D = [0;L]3 ⊆ R3. Inside this cube we have electromagnetic radiation that obeys Maxwell's equations bouncing around. The equipartition theorem of thermodynamics says that every wave which has con- stant spatial structure with oscillating amplitude, i.e.
    [Show full text]
  • Heat Kernel Analysis on Graphs
    Heat Kernel Analysis On Graphs Xiao Bai Submitted for the degree of Doctor of Philosophy Department of Computer Science June 7, 2007 Abstract In this thesis we aim to develop a framework for graph characterization by com- bining the methods from spectral graph theory and manifold learning theory. The algorithms are applied to graph clustering, graph matching and object recogni- tion. Spectral graph theory has been widely applied in areas such as image recog- nition, image segmentation, motion tracking, image matching and etc. The heat kernel is an important component of spectral graph theory since it can be viewed as describing the flow of information across the edges of the graph with time. Our first contribution is to investigate how to extract useful and stable invari- ants from the graph heat kernel as a means of clustering graphs. The best set of invariants are the heat kernel trace, the zeta function and its derivative at the origin. We also study heat content invariants. The polynomial co-efficients can be computed from the Laplacian eigensystem. Graph clustering is performed by applying principal components analysis to vectors constructed from the invari- ants or simply based on the unitary features extracted from the graph heat kernel. We experiment with the algorithms on the COIL and Oxford-Caltech databases. We further investigate the heat kernel as a means of graph embedding. The second contribution of the thesis is the introduction of two graph embedding methods. The first of these uses the Euclidean distance between graph nodes. To do this we equate the spectral and parametric forms of the heat kernel to com- i pute an approximate Euclidean distance between nodes.
    [Show full text]
  • Spectral Geometry for Structural Pattern Recognition
    Spectral Geometry for Structural Pattern Recognition HEWAYDA EL GHAWALBY Ph.D. Thesis This thesis is submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy. Department of Computer Science United Kingdom May 2011 Abstract Graphs are used pervasively in computer science as representations of data with a network or relational structure, where the graph structure provides a flexible representation such that there is no fixed dimensionality for objects. However, the analysis of data in this form has proved an elusive problem; for instance, it suffers from the robustness to structural noise. One way to circumvent this problem is to embed the nodes of a graph in a vector space and to study the properties of the point distribution that results from the embedding. This is a problem that arises in a number of areas including manifold learning theory and graph-drawing. In this thesis, our first contribution is to investigate the heat kernel embed- ding as a route to computing geometric characterisations of graphs. The reason for turning to the heat kernel is that it encapsulates information concerning the distribution of path lengths and hence node affinities on the graph. The heat ker- nel of the graph is found by exponentiating the Laplacian eigensystem over time. The matrix of embedding co-ordinates for the nodes of the graph is obtained by performing a Young-Householder decomposition on the heat kernel. Once the embedding of its nodes is to hand we proceed to characterise a graph in a geometric manner. With the embeddings to hand, we establish a graph character- ization based on differential geometry by computing sets of curvatures associated ii Abstract iii with the graph nodes, edges and triangular faces.
    [Show full text]
  • Inverse Spectral Problems in Riemannian Geometry
    Inverse Spectral Problems in Riemannian Geometry Peter A. Perry Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506-0027, U.S.A. 1 Introduction Over twenty years ago, Marc Kac posed what is arguably one of the simplest inverse problems in pure mathematics: "Can one hear the shape of a drum?" [19]. Mathematically, the question is formulated as follows. Let /2 be a simply connected, plane domain (the drumhead) bounded by a smooth curve 7, and consider the wave equation on /2 with Dirichlet boundary condition on 7 (the drumhead is clamped at the boundary): Au(z,t) = ~utt(x,t) in/2, u(z, t) = 0 on 7. The function u(z,t) is the displacement of the drumhead, as it vibrates, at position z and time t. Looking for solutions of the form u(z, t) = Re ei~tv(z) (normal modes) leads to an eigenvalue problem for the Dirichlet Laplacian on B: v(x) = 0 on 7 (1) where A = ~2/c2. We write the infinite sequence of Dirichlet eigenvalues for this problem as {A,(/2)}n=l,c¢ or simply { A-},~=1 co it" the choice of domain/2 is clear in context. Kac's question means the following: is it possible to distinguish "drums" /21 and/22 with distinct (modulo isometrics) bounding curves 71 and 72, simply by "hearing" all of the eigenvalues of the Dirichlet Laplacian? Another way of asking the question is this. What is the geometric content of the eigenvalues of the Laplacian? Is there sufficient geometric information to determine the bounding curve 7 uniquely? In what follows we will call two domains isospectral if all of their Dirichlet eigenvalues are the same.
    [Show full text]
  • Isospectral Towers of Riemannian Manifolds
    New York Journal of Mathematics New York J. Math. 18 (2012) 451{461. Isospectral towers of Riemannian manifolds Benjamin Linowitz Abstract. In this paper we construct, for n ≥ 2, arbitrarily large fam- ilies of infinite towers of compact, orientable Riemannian n-manifolds which are isospectral but not isometric at each stage. In dimensions two and three, the towers produced consist of hyperbolic 2-manifolds and hyperbolic 3-manifolds, and in these cases we show that the isospectral towers do not arise from Sunada's method. Contents 1. Introduction 451 2. Genera of quaternion orders 453 3. Arithmetic groups derived from quaternion algebras 454 4. Isospectral towers and chains of quaternion orders 454 5. Proof of Theorem 4.1 456 5.1. Orders in split quaternion algebras over local fields 456 5.2. Proof of Theorem 4.1 457 6. The Sunada construction 458 References 461 1. Introduction Let M be a closed Riemannian n-manifold. The eigenvalues of the La- place{Beltrami operator acting on the space L2(M) form a discrete sequence of nonnegative real numbers in which each value occurs with a finite mul- tiplicity. This collection of eigenvalues is called the spectrum of M, and two Riemannian n-manifolds are said to be isospectral if their spectra coin- cide. Inverse spectral geometry asks the extent to which the geometry and topology of M is determined by its spectrum. Whereas volume and scalar curvature are spectral invariants, the isometry class is not. Although there is a long history of constructing Riemannian manifolds which are isospectral but not isometric, we restrict our discussion to those constructions most Received February 4, 2012.
    [Show full text]
  • Notes on the Spectral Theorem
    Math 108b: Notes on the Spectral Theorem From section 6.3, we know that every linear operator T on a finite dimensional inner prod- uct space V has an adjoint. (T ∗ is defined as the unique linear operator on V such that hT (x), yi = hx, T ∗(y)i for every x, y ∈ V – see Theroems 6.8 and 6.9.) When V is infinite dimensional, the adjoint T ∗ may or may not exist. One useful fact (Theorem 6.10) is that if β is an orthonormal basis for a finite dimen- ∗ ∗ sional inner product space V , then [T ]β = [T ]β. That is, the matrix representation of the operator T ∗ is equal to the complex conjugate of the matrix representation for T . For a general vector space V , and a linear operator T , we have already asked the ques- tion “when is there a basis of V consisting only of eigenvectors of T ?” – this is exactly when T is diagonalizable. Now, for an inner product space V , we know how to check whether vec- tors are orthogonal, and we know how to define the norms of vectors, so we can ask “when is there an orthonormal basis of V consisting only of eigenvectors of T ?” Clearly, if there is such a basis, T is diagonalizable – and moreover, eigenvectors with distinct eigenvalues must be orthogonal. Definitions Let V be an inner product space. Let T ∈ L(V ). (a) T is normal if T ∗T = TT ∗ (b) T is self-adjoint if T ∗ = T For the next two definitions, assume V is finite-dimensional: Then, (c) T is unitary if F = C and kT (x)k = kxk for every x ∈ V (d) T is orthogonal if F = R and kT (x)k = kxk for every x ∈ V Notes 1.
    [Show full text]