Calculation of Heat Determinant Coefficients for Scalar Laplace type Operators

Transcription

1 Calculation of Heat Determinant Coefficients for Scalar Laplace type Operators by Benjamin Jerome Buckman Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science in Mathematics with Specialty in Analysis New Mexico Institute of Mining and Technology Socorro, New Mexico August, 04

2 ABSTRACT In spectral geometry, one would like to know how much we can distinguish two manifolds by the spectrum of a differential operator. Heat invariants, such as the heat trace and heat content have been studied and isospectral manifolds have been found. We introduce a new heat invariant called the heat determinant which is a step foreword in the study of non-spectral invariants on manifolds. The heat determinant is not a pure spectral invariant in that it does not depend only on the eigenvalues of an operator, but also the eigenfunctions of that operator as well. We find the asymptotic expansion in t of this new invariant and calculate the first 3 terms of the heat determinant for the scalar Laplacian on manifolds without boundary. Keywords: Heat invariants; Heat kernel; Asymptotic expansions; Spectral geometry; Heat determinant

3 CONTENTS. INTRODUCTION. BACKGROUND INFORMATION 3. Differential Geometry Basic Definitions Riemannian Metric Tensors Covariant Derivative Curvature Geodesics Synge Function Covariant Taylor Series Laplace Type Operators Heat Kernel Heat Invariants Heat Trace Heat Content Modified Heat Determinant HEAT DETERMINANT 0 3. Definition of Heat Determinant Calculation of Heat Determinant Mixed Derivative of Heat Kernel Heat Determinant Asymptotics Heat Determinant Asymptotics for Scalar Operators Coefficients of Heat Determinant Asymptotic Expansion CONCLUSION 36 ii

4 5. APPENDIX Gaussian Integrals Evaluation of Coincidence Limits REFERENCES 44 iii

5 This thesis is accepted on behalf of the faculty of the Institute by the following committee: Ivan Avramidi, Advisor I release this document to the New Mexico Institute of Mining and Technology. Benjamin Jerome Buckman Date

6 CHAPTER INTRODUCTION In 966, Marc Kac popularized the question Can you hear the sound of a drum? 4]. In other words, can one determine the shape of a drum based on the frequencies one can hear? The area of spectral geometry was created to answer this question. We take a manifold and find the spectrum of some differential operator, such as the Laplacian, on this manifold. It is obvious that if you have a large drum and a small drum, the frequencies will be different. In spectral geometry, we want to answer the question how much does the spectrum of an operator determine the geometry of our manifold. About the same time Kac popularized the question, two different manifolds distinct 6-dimensional tori were found to have the same eigenvalues 5]. The problem in -dimensions was not answered till 99 ]. Thus one cannot hear the shape of a drum. Spectral geometry then became the question of how much we can infer of the geometry of an object from the spectrum of an operator. Mathematicians then turned to spectral invariants, objects that are invariant on a manifold that depend on the spectrum of an operator. In practice, it is rather difficult to calculate the eigenvalues of a differential operator on a manifold, thus we study asymptotic regimes. One of the most important operators on a manifold is the Laplacian,. It is the most natural second order elliptic partial differential operator on a manifold and it is seen in many important partial differential equations such as the wave and diffusion equations. The wave equations describes how waves propagate through space, while the diffusion equation describes how some material diffuses through space, such as particles or energy. For this paper, we will focus on the heat equation which describes how heat diffuses through a system. The heat equation is given by t f t, x = 0,. which we must give an initial condition f 0, x = gx,. and appropriate boundary conditions if present. The fundamental solution of the heat equation is called the heat kernel. The heat kernel, itself, has many applications to physics and mathematics, more specifically quantum gravity and spectral geometry.

7 From the heat kernel, we can define many invariants. Some spectral invariants that have been studied already include the heat trace and the heat content. In this paper we introduce a non-spectral invariant called the heat determinant. This new invariant is non-spectral since it depends on the eigenvalues and the eigenfunctions of our operator. We calculate the first 3 terms in the asymptotic expansion of the heat determinant. This paper is organized as follows: Chapter contains all the background information we need, focusing on differential geometry. It will introduce all the notation and objects we will be using throughout this paper. We also mention other heat invariants. In chapter 3, we define the heat determinant on vector bundles. We then go through calculation of the heat determinant, then calculating its asymptotics. In section 3.4, we calculate the first 3 coefficients in our asymptotic expansion. In chapter 4, we have our conclusion where we summarize our results. In chapter 5, we have our appendix where we have put many of our technical calculations.

8 CHAPTER BACKGROUND INFORMATION. Differential Geometry In this section, we will have a brief overview of differential geometry and the tools which we will be using. For a more in depth introduction, see 3], 8], or 4]... Basic Definitions In this section, we will give the standard definitions of the mathematical objects we will be using throughout. Definition. An n-dimensional manifold without bounday, M, is a Hausdorff, second countable, topological space such that for any point p in M, there exists a homeomorphism ϕ U : U V, where U is an open neighborhood of p and V is an open subset of R n. We call the ordered pair U, ϕ U a coordinate patch and we define the local coordinate of a point p M as ϕ U p V R n. A manifold is called a differential manifold if for every pair of coordinate patches, U, ϕ U and U, ϕ U with U U =, the new map φ = ϕ U ϕu : V R n R n is differentiable. We will only deal smooth manifolds, that is, φ is smooth i.e. infinitely differentiable for all pairs of coordinate patches with non-empty intersections. All manifolds herein are smooth manifolds of dimension n. Definition. A vector also called tangent vector or contravariant vector at a point p in a coordinate patch U, x U on a manifold M is an element of R n denoted as X p = X µ U = XU,, XU n such that for any other coordinate patch U, x U with p U, then X µ n µ ] x U U = x ν= U ν p XU ν.. 3

9 A vector can also be represented as the differential operator ] X p =,. n ν= XU ν xu ν where / xu ν is the coordinate basis of U in local coordinates. Definition. The Jacobian matrix between two coordinate patches given by U, x U and U, x U, is the matrix of partial derivatives µ x U xu ν.3 p and the Jacobian is J = det µ x U xu ν..4 Suppose we have another smooth manifold, B, and a smooth mapping π : B M. Then we define the triple B, π, M as a bundle where B is the bundle space and π is the projection. The we define a fiber as the image of π p, where p M. A section of a bundle is a map s : M B such that sp π p. If we let B be a vector space, then we call our bundle a vector bundle. We define the tangent space of M at a point p as the vector space containing all tangent vectors to M at p and is denoted as T p M. We define tangent bundle as the set of all tangent spaces at every point of M and it is denoted as TM. Then we define a vector field as a smooth mapping X : M TM which can also be represented in local coordinates as Xp = n X ν p x ν= ν..5 Definition. A covector also called a cotangent or covariant vector is a linear functional α p : T p M R and is referred to as the dual of a vector. Thus the dual of T p M is the set of all linear functionals on T p M and is called the cotangent space at the point p and is denoted as Tp M. Similarly, the cotangent bundle is the set of all cotangent spaces at every point of M and is denoted as T M. Then we define a covector field also called a -form as a smooth mapping α : M T M. We also have a covector bundle which is the dual to the vector bundle. We will henceforth use the notation µ = x µ. The dual to the coordinate basis µ is denoted as dx µ and satisfies the condition dx µ ν = ν dx µ = δ µ ν,.6 4

10 where δ µ ν = { if µ = ν 0 if µ = ν.7 is the Kronecker delta. We will be using Einstein summation notation throughout this paper, thus we sum over repeated indices i.e. β µ α µ = µ β µ α µ... Riemannian Metric A Riemannian metric is the smooth assignment of a positive definite inner product in T p M to p M. Definition. An inner product on a vector space v is defined as the mapping, : V V C that satisfies the properties that for all x, y, z V and α, β C,. x, y = y, x,. αx + βy, z = α x, z + β y, z, 3. x, x 0, 4. x, x = 0 implies x = 0. Then inner product defines a norm denoted as v = v, v. The Riemannian metric is denoted as g µν and it is a function of x on M. The metric is a positive definite matrix. The inverse of the metric is denoted as g µν = g µν. We will also denote the determinant of the metric as g = det g µν. Thus the inner product defined on T p M is given by the metric as u, v = g µν u µ v ν..8 The inverse defines an inner product on T p M. Using g µν and g µν, we are able to raise and lower indices, thus each vector has a corresponding covector and vise-versa given by v µ = g µν v ν or α µ = g µν α ν..9 Definition. Suppose we have a smooth curve on a manifold M parametrized by a variable τ 0, t]. Then xτ = x τ,, x n τ is the trajectory of the curve in M. Then the µ th component of the tangent vector of the curve is dxµ τ dτ. Therefore the arclength of the curve from τ = 0 to τ = t is s = t 0 dτ dxτ dτ = 5 t 0 dτ g µν dx µ dτ dx ν dτ..0

11 The standard Euclidean volume form denoted as dx = dx dx n is not invariant under coordinate transformation. Thus we introduce the Riemannian volume form which is invariant. Definition. The Riemannian volume element is. d volx = g xdx....3 Tensors Definition. A tensor of type p,q of a vector space V is a multilinear map T : V } {{ V } p times V } {{ V } R. q times and is denoted as where each index goes from to n. T µ µ µ p ν ν ν q,.3 A tensor is invariant under coordinate changes with every upper index transforming as a vector and every lower index transforming as a covector. Then a tensor field is the smooth assignment of tensor to every point of M and is denoted as T µ µ µ p ν ν ν q x..4 Suppose T is a tensor of type,0 or, or 0,, then we call T a matrix. Then we denote the trace of the matrix T as tr T = g µν T µν = T µ µ..5 Definition. Let S q be the group of permutations of the set Z q permutation ϕ : Z q Z q is represented by q ϕ ϕq = {,,, q}. A..6 An elementary permutation is a permutation that exchanges the order of only two elements. We say a permutation is even odd if it is the product of an even odd number of elementary permutations. We define the function { if ϕ is even signϕ = if ϕ is odd..7 6

12 Using the above definition, we define the anti-symmetrization and symmetrization of tensors: Definition. Let T be a tensor of type 0,q. the symmetrization of the indices of tensor T is given by T ν ν ν q = q! ϕ S q T νϕ ν ϕ ν ϕq..8 Secondly, we define the anti-symmetrization of tensor T as T ν ν ν q ] = q! ϕ S q signϕt νϕ ν ϕ ν ϕq..9 We will use vertical lines to denote omission of an index from symmetrization and antisymmetrization, i.e. T ν ν ν j ν j ν j+ ν q ] = q! ϕ S q signϕt νϕ ν ϕ ν ϕj ν j ν ϕj+ ν ϕq..0 The Levi-Civita symbol is not a tensor, but is what we call a tensor density and it will be used often. The Levi-Civita symbol is ε µ µ µ n = ε µ µ µ n if {µ, µ,, µ n } is an even perm. of {,,, n} = if {µ, µ,, µ n } is an odd perm. of {,,, n}. 0 otherwise. Throughout this paper, we will use the Levi-Civita density, which is just the Levi- Civita symbol scaled by the metric. The density is invariant and indices can be raised or lowered by the metric. The density will be denoted as E µ µ µ n = g εµ µ µ n and E µ µ µ n = g ε µ µ µ n.. The Levi-Civita symbol is most helpful in writing the determinant of a matrix: deta µ ν = ε µ µ n A µ A n µ n = ε ν ν n A ν A ν n n = n! εµ µ n ε ν ν n A ν µ A ν n µn..3 We note that the Levi-Civita symbol satisfies ε µ µ n A ν µ A ν n µn = ε ν ν n det A µ ν,.4 7

13 and ε µ µ k α α n k ε ν ν k α α n k = n k! δ µ µ k ν ν k,.5 where These will be used in the following lemma. δ µ µ k ν ν k = k! δ µ ν δ µ k ν k ]..6 Lemma. Suppose we have a matrix A µ ν with its inverse A µ ν denoted as B ν µ, and we define the object F µ µ k ν ν k = n k! εµ µ k α α n k ε ν ν k β β n k A β α A β n k αn k,.7 then F µ µ k ν ν k = k!b µ ν B µ k] νk deta..8 Proof. Let us multiply.7 by k A s, then we obtain A ν λ A ν k λk F µ µ k ν ν k = n k! εµ µ k α α n k ε ν ν k β β n k A ν λ A ν k λk A β α A β n k αn k.9 Now, by multiplying by k B s, we have = n k! εµ µ k α α n k ε λ λ k α α n k deta.30 =δ µ µ k λ λ k deta..3 B λ κ B λ k κk A ν λ A ν k λk F µ µ k ν ν k =δ ν κ δ ν k κ k F µ µ k ν ν k =F µ µ k κ κ k..3 Therefore F µ µ k κ κ k = B λ κ B λ k κk δ µ µ k λ λ k deta = k!b µ κ B µ k] κk deta..33 8

14 ..4 Covariant Derivative Partial derivatives are not invariant under a change of coordinates, therefore we must define an invariant derivative. First we need the affine connection which relates the tangent vectors at two different points on a manifold. The affine connection that is torsion-free and compatible with the metric is called the Levi- Civita connection and is denoted by the Christoffel symbols which are given by Γ α µν = gαβ µ g βν + ν g µβ β g µν..34 Then the covariant derivatives of a vector field v and a covector field α are ν v µ = ν v µ + Γ µ βνv β and ν α µ = ν α µ Γ β µνα β..35 We also have covariant derivative of a tensor: α T µ µ µ p ν ν ν q = α T µ µ µ p ν ν ν q Γ β ν αt µ µ µ p βν ν q + Γ µ βα T βµ µ p ν ν ν q + + Γ µ p βα T µ µ β ν ν ν q Γ β ν q αt µ µ µ p ν ν β..36 We also consider a vector bundle, V, on our manifold, M. Let A µ be a connection on this vector bundle, then the covariant derivative of a section ϕ of the bundle V is given by µ ϕ = µ ϕ + A µ ϕ Curvature The Riemann curvature tensor R µ ναβ is found by finding the commutator of covariant derivatives evaluated on a vector. So we get α, β ] v µ = R µ ναβv ν.38 with R µ νγδ = γ Γ µ νδ δ Γ µ νγ + Γ µ βγγ β νδ Γ µ βδγ β νγ..39 We can also evaluate the commutator of covariant derivatives on a covector and we have γ, δ ] α µ = R ν µγδα ν..40 get Lastly, we can evaluate the commutator on a tensor of type p, q and we γ, δ ] T µ µ µ p ν ν ν q = p R µ a βγδ T µ µ a βµ a+ µ p ν ν ν q a= 9 q R β ν b γδt µ µ µ p ν ν b βν b+ ν q. b=.4

15 We can raise and lower indices on the Riemann tensor just like any tensor. The Riemann tensor is anti symmetric in the first two indices i.e. R µνγδ = R νµγδ, anti symmetric in the last two indices i.e. R µνγδ = R µνδγ, and is symmetric in the exchanging of these pairs i.e. R µνγδ = R γδµν. By contracting over indices, we can define new tensors. Definition. The Ricci tensor is given by and the scalar curvature is given by R µν = R γ µγν.4 R = g µν R µν = R µ µ..43 The commutator of covariant derivatives of a section, ϕ, of a vector bundle, V, defines the curvature -form, R, of the bundle connection, A, µ, ν ] ϕ = Rµν ϕ,.44 where R µν = µ A ν ν A µ + A µ, A ν ] Geodesics Definition. A geodesic is a curve xτ on a manifold M such that the following equation holds: dx ν dx dτ µ ν = d x µ dτ dτ + dx ν dx γ Γµ νγ dτ dτ = 0.46 for some parametrization τ which we call the affine parameter. A geodesic is a curve for which the tangent vector to the curve transported along the curve remains the tangent vector and is referred to as the shortest curve between two points. Any two points on a manifold M that are sufficiently close can be connected by a single geodesic. The distance along the geodesic between two points x and x is called the geodesic distance and is denoted as dx, x...7 Synge Function Before we define the Synge function, we need to introduce the injectivity radius of a manifold. 0

16 Definition. Let x M, then we define the ball of radius r as B r x = { x M dx, x < r }..47 Definition. Let x, x U M. We say x and x are conjugate in U if they are connected by more than one geodesic. Definition. Let r inj x be the largest radius of the ball B r x such x B r x, x and x are not conjugate in B r x. In other words, r inj x is the largest ball around x such that all points are connected to x by a single geodesic. We call r inj x the injectivity radius of the point x. Definition. The injectivity radius of the manifold, M, is given by Γ inj M = inf x M r injx..48 Thus if r < Γ inj, then x M, B r x will not contain conjugate points. Definition. The Synge function also called a world function is a bi-scalar function and is defined as half the square geodesic distance between points x and x on M. Thus σx, x = d x, x = t t 0 dτ g µν dx µ dτ dx ν dτ..49 We note σ is multivalued for conjugate points. We will consider σ as single valued when, along the shortest geodesic between x and x, dx, x < Γ inj. We will then use this distance along this shortest geodesic as dx, x in.49. Thus σ is only defined locally. Before we continue further, we will mention some of the notation we will be using. The coincidence limit of a bi-scalar function is denoted as f x, x ] = lim f x, x..50 x x For the Synge function, we will use indices to denote covariant derivatives: σ µ = µ σx, x and σ µ = µ σx, x,.5 where µ is the covariant derivative with respect to the x coordinate and µ is the covariant derivative with respect to the x coordinate. Thus we can have multiple indices indicating multiple derivatives i.e. σ µ ν = σ µν = µ ν σ = ν µ σ.

17 The Synge function is very important in that it completely describes the local geometry near x and x. The Synge function has the following properties: σ µ σ µ =σ, σ ν σ ν =σ,.5 σ µ σ µ ν =σ ν, σ µ σ µ ν =σ ν,.53 σ ν σ ν µ =σ µ, σ µ σ µ ν =σ ν..54 We will denote the inverse matrix of σ µν as γ µν, the inverse matrix of σ µν as γ µν, and the inverse matrix of σ µ ν as γµ ν. Thus it is easy to see that γ matrices satisfy γ µν σ να = δ µ α, γ µν σ ν α = δ µ α,.55 σ µ γ µν = σ ν, σ µ γ µν = σ ν,.56 σ ν γ ν µ = σ µ, σ µ γ µ ν = σ ν..57 See Section 5. for more on the Synge function, its derivatives and their coincidence limits which will be used extensively. We introduce the Van Fleck-Morette determinant which is defined as x, x = g xg x det σ µν..58 Again, see Section 5. for information on the derivatives and the coincidence limits of. We will use the following change of variables later: These change of coordinates satisfy the following: ξ µ = σµ t and ξ ν = σν t..59 ξ = ξ µ ξ µ = ξ ν ξ ν = σ t,.60 σ ν µξ µ = ξ ν and σ µν ξ ν = ξ µ..6 Then the standard partial derivatives transform as x µ = σν t µ ξ ν and = tγ µ ξ ν ν x µ..6 Thus we have σ µ x µ = ξν,.63 ξ ν

18 and the volume element becomes dvolx = g x dx = t n x, x g x dξ..64 We also introduce the Gaussian average over the variable ξ a n f ξ a = dξ g x exp aξ f ξ,.65 π R n where a is a positive constant. In particular, we have ξ µ ξ µ ξ µ k+ a = 0,.66 ξ µ ξ µ ξ µ k a = k! a k k! gµ µ g µ k µ k..67 For more on the Gaussian integral and Gaussian average, see section Covariant Taylor Series We consider a section, ϕ, of a vector bundle, V, on the manifold M. To expand ϕ in covariant Taylor series, we will transport ϕ from x to x, then expand it in a Taylor series along the geodesic, then transport it back to x. We first introduce the parallel transport operator Px, x, which parallel transports a field from the point x to x along the connecting geodesic. Thus, the parallel transport operator is defined as σ µ µ P = 0,.68 with P] =. By taking derivatives of the above equation and symmetrizing, it is easy to see that ] µ µk P = 0,.69 with the same properties holding for primed derivatives and inverse of P given by P x, x = Px, x. Therefore, let us define k=0 ϕ = P ϕ..70 We expand this in a Taylor series along the geodesic using our affine parameter, τ. We find d ϕ = k k! τk dτ k ϕ..7 τ=0 3

19 We know that d dτ = Uµ µ, where U µ = dxµ dτ. Noting that µ ϕ = µ ϕ, and that U µ µ U ν = 0 since it is just the geodesic equation, we have ϕ = k=0 k! τk U µ U µ k µ µk P ϕ τ=0..7 Noting that τu µ 0 = σ µ as derived in Section 5., and that evaluating P and its derivatives at τ = 0 is the same as taking the coincidence limit, we find ϕ = k=0 k σ µ σ µ k k! ] µ µk ϕ..73 Therefore, we have the covariant Taylor expansion of a field ϕ given by ϕ = P k=0 k σ µ σ µ k k! ] µ µk ϕ..74. Laplace Type Operators For this paper, we will only be looking at second-order elliptic partial differential operators, more specifically, the scalar Laplacian acting on manifolds without boundary. However, the heat determinant we introduce later can be applied any second-order elliptic partial differential operator. Definition. Let H = L V be the Hilbert space complete inner product space of square integrable sections of our vector bundle V. The inner-product on H is defined as ϕ, ψ = dvolx ϕ, ψ,.75 where, is the inner product in the fiber of our bundle V. M Suppose we have a second order partial differential operator given by L = a µν µ ν + b µ µ + c,.76 where a, b, and c could all be functions of x. Then we denote the leading symbol as σ L ξ, x = a µν ξ µ ξ ν. The operator L is elliptic if ξ = 0 and x M, σ L ξ, x C ξ for some C > 0. i.e. where ξ = δ µν ξ µ ξ ν. a µν x ξ µ ξ ν C ξ ξ = 0 and x M..77 Let C V be the space of smooth functions on V. 4

20 Definition. A Laplace type operator is an elliptic operator L : C V C V of the form L = + Q.78 where Q is a smooth endomorphism of the bundle V and is the standard Laplacian = g µν µ ν..79 The adjoint of an operator L is denoted as L which satisfies f, Lg = L f, g.80 for all f and g in H. An operator is self-adjoint if L = L and the domains of L and L are identical. The Laplace type operators are symmetric i.e. f, Lg = L f, g. Laplace type operators have self-adjoint extensions, but they, themselves, are not selfadjoint. We will not distinguish between L and its self-adjoint extension. We only deal with self-adjoint elliptic operators on compact manifolds without boundary, since they have a countable set of eigenvalues and eigenfunctions. Recall that if λ i is an eigenvalue with the corresponding eigenfunction ϕ i, then they satisfy the equation Lϕ i = λ i ϕ i..8 If the set of eigenfunctions forms a complete orthonormal basis on our Hilbert space, then we can spectrally decompose a function of the operator L: f Lφ = f λ k ϕ k, φ ϕ k..8 k=0 where φ is a section of our vector bundle. This makes life easy when dealing with elliptic operators. We define the trace of a function of an operator as the sum of the function evaluated at our eigenvalues, Tr f L = f λ k..83 k=0 For a more indepth discussion on operators and spectral decomposition, see 6]. 5

21 .. Heat Kernel As we have already mentioned, we will only study compact manifolds without boundary. The heat kernel of an operator L is denoted as Ut, x, x and satisfies the heat equation with the initial condition t + L Ut, x, x = 0,.84 U0, x, x = δx, x,.85 where δx, x = g 4 xg 4 x δx x is the covariant δ function. The operator L acts on unprimed coordinates. The heat kernel gives us the general solution to to an arbitrary initial condition. That is, the solution of the initial value problem t + L f t, x = 0,.86 f 0, x = hx,.87 is given by f t, x = M dvolx Ut, x, x hx..88 We can spectrally decompose the heat kernel to Ut, x, x = exp tλ k ϕ k xϕ k x..89 k=0.3 Heat Invariants In this section, we discuss a few heat invariants. We will assume L is the general scalar Laplace type operator where Q is a non-constant smooth endomorphism. L = + Q,.90 6

22 .3. Heat Trace One of the most important heat invariants is the heat trace. It has been studied extensively ]. The heat trace is given by the trace of the heat semigroup exp tl, Θt = Tr exp tl = e tλ k..9 k= This is a pure spectral invariant, the asymptotics of which play an important role in spectral geometry and mathematical physics. It is well known that as t 0, Θt 4πt n k A k! k t k,.9 k=0 where A k are called the heat trace invariants. They are given by integral of local invariants, more specifically, the first two are A 0 = volm,.93 A = dvol Q 6 R..94 M.3. Heat Content Another heat invariant is the heat content and is given by the integral of the heat kernel of the manifold. For scalar operators, the heat content is defined by Πt = dvolx dvolx Ut, x, x = e tλ k Φ k,.95 k= where M M Φ k = M dvol ϕ k..96 From.89, we find that dvolx Ut; x, x = exp tl x..97 M Thus expanding the heat content as t gives us Πt k=0 k Π k! k t k,.98 7

23 where Π 0 = volm,.99 Π = dvol Q,.00 Π k = M We note that if Q = 0, then Πt = volm. M dvol Q + Q k Q, k Modified Heat Determinant We introduce a modified heat determinant which we will not use since it vanishes for manifolds without boundary. We define a matrix P as P µν t; x, x = µ ν Ut; x, x = e tλ l µ ϕ l x ν ϕ l x. l= Thus we can define the modified heat determinant as Kt = dx dx det P µν t; x, x M M = n! M M.0 dx dx ε µ µ n ε ν ν n P µ ν P µn ν n..03 We define a new object Ẽ as Ẽ k k n = = M M dx ε µ µ n µ ϕ k µn ϕ kn dϕ k dϕ kn..04 We note that dϕ k dϕ kn = dϕ k dϕ k dϕ kn, thus using Stokes theorem, we have Ẽ k k n = ϕ k dϕ k dϕ kn..05 M Thus Ẽ = 0 if our manifold has no boundary. 8

24 We can write K as Kt = dx dx ε µ µ n ε ν ν n n! M M {. k n = k = e tλ k µ ϕ k x ν ϕ k x } e tλ kn µn ϕ kn x ν n ϕ kn x,.06 then Kt = n! = k,,k n = e tλ k k k n e tλ k + +λ k n Ẽ k k n Ẽ k k n + +λ k n Ẽ k k n Ẽ k k n..07 From.05, it is clear that for manifolds without boundary K = 0. However if the boundary of our manifold is not empty, then the modified heat determinant is non-zero and we can study this new heat invariant that depends on the eigenvalues and eigenfunctions of our operator L. However, this only works for a scalar operator, as it is not invariant for elliptic second-order differential operators on vector bundles. 9

25 CHAPTER 3 HEAT DETERMINANT In this section, we will define the heat determinant of an elliptic second order partial differential operator on a vector bundle, then look more specifically at the heat determinant for scalar Laplacian. 3. Definition of Heat Determinant Define the tensor P as P µν t; x, x = tr U t; x, x µ ν Ut; x, x, 3. where Ut; x, x is the heat kernel of an operator L. We note that the trace here is over the fiber of our vector bundle, thus P µν is a matrix with scalar entries. We also not that if L is self-adjoint, then U t, x, x = Ut, x, x. Then the heat determinant is defined as Kt = dx dx detp µν t, x, x, 3. M M which can also be written as Kt = dx dx ε µ µ n ε ν ν n P n! µ ν t, x, x P µn ν n t, x, x. 3.3 M M Let ϕ k be the eigenfunction corresponding to the eigenvalue λ k of the operator L. Then we define f kl x = ϕ k, ϕ l, 3.4 then we have Now define ϕ k, ϕ l = M dvolx f kl x = δ kl. 3.5 B kl µ = ϕ k, µ ϕ l, 3.6 0

26 which then defines the one-forms B kl = B kl µ dx µ = ϕ k, µ ϕ l dx µ = ϕ k, Dϕ l. 3.7 Lastly, we define E l l n k k n = = M M Thus we can write the tensor P as { P µν t; x, x = tr Then Kt = = = k,l= k= dx ε µ µ n ϕk, µ ϕ l ϕkn, µn ϕ ln B k l B kn l n. e tλ k ϕ k x ϕ k x l= e tλ l µ ϕ l x ν ϕ l x e tλ k+λ l ϕ k x, µ ϕ l x ϕ k x, ν ϕ l x e tλ k+λ l B kl µ xbkl ν x. k,l= n! M M dx dx ε µ µ n ε ν ν n k,l= e tλ k+λ l B kl µ xb kl ν x } = n!. k l k n l n = k,l= M M e tλ k+λ l B kl µn xb kl ν n x dx dx ε µ µ n ε ν ν n e tλ k +λ l Bk l µ xb k l ν x. e tλ kn +λ ln B kn l n µ n xb k n l n ν n x 3. = n! e tλ k +λ l + +λ k n +λ ln E l l n k k n E l l n k k n. 3. k l k n l n =

27 3. Calculation of Heat Determinant Let s fix a point x. Then in a ball B r x of radius r < Γ inj, the heat kernel for an arbitrary elliptic second order differential operator is given by Ut; x, x = 4πt { exp n/ σx, x t } x, x Px, x Ωt; x, x, 3.3 where x, x is the Van-Fleck-Morette determinant.58, Px, x is the parallel transport operator, and Ωt, x, x is the transport function. It is well known that at t 0, Ω as the asymptotic expansion given by Ωt, x, x k=0 t k a k! k x, x, 3.4 where a n x, x are the heat kernel coefficients which have been calculated see ] and sources therein. In this section, we calculate the mixed derivative of the heat kernel and find a general formula for the det P. 3.. Mixed Derivative of Heat Kernel We look at the mixed derivative of the heat kernel first for vector valued operators. For an arbitrary vector-valued second order differential operator, let us define Ψt; x, x = x, x Px, x Ωt; x, x. 3.5 Then Ut; x, x = 4πt exp n/ σx, x t Taking two covariant derivatives of this, we have µ ν U = 4πt Then we get exp n/ σ t 4t σ µσ ν Ψ t Ψt; x, x. 3.6 σ µν Ψ + σ µ Ψ ν + σ ν Ψ µ + Ψ µν. 3.7 P µν = tr U t, x, x µ ν Ut, x, x = 4πt n exp σ t t Y µν, 3.8

28 where Y µν = t σ µσ ν tr Ψ Ψ σ µν tr Ψ Ψ + σ µ tr Ψ Ψ ν + σ ν tr Ψ Ψ µ + t tr Ψ Ψ µν. 3.9 So if we define the following: S µν = σ µ σ ν tr Ψ Ψ, 3.0 V µν = σ µ tr Ψ Ψ ν, 3. W µν = σ ν tr Ψ Ψ µ, 3. Z µν = σ µν tr Ψ Ψ + t tr Ψ Ψ µν, 3.3 then Y µν = Z µν + W µν + V µν + t S µν. 3.4 Since S, V, W, and Z are defined this way, they satisfy the following relations: S µ ν S µ ] ν ] = V µ ν V µ ] ν ] = W µ ν W µ ] ν ] = S µ ν V µ ] ν ] = S µ ν W µ ] ν ] = Thus where det P µν = 4πt n t exp σ n det Y t µν, 3.6 det Y µν = { n! εµ µ n ε ν ν n Z µ ν Z µ n ν n + n W µ ν + V µ ν + t S µ ν Z µ ν Z µ n ν n } + n n W µ ν V µ ν Z µ 3 ν 3 Z µ n ν n. 3.7 So det P µν x, x = g xg x exp n t n n 4πt t σx, x Ht; x, x, 3.8 where Ht; x, x = g x g x det Y µν t; x, x

29 is Therefore, our heat determinant is given by 3., which in all of its glory Kt = t n 4πt n M M dx dx exp n t σx, x det Y µν t; x, x Let r be a real number less than the injectivity radius of the manifold, Γ inj M. Then we note that Kt = K t + K t, 3.3 where K t = dx dx exp n t n 4πt n t σx, x det Y µν t; x, x, 3.3 M B r x K t = dx dx det P µν t; x, x M M B r x By standard elliptic estimates, it has been shown 3] that x M B r x and 0 < t < that the heat kernel has the estimate Ut; x, x C t n exp r, t where C is a constant. From the heat equation, we note that each spatial derivative has the dimension of t. Thus we have an estimate for 3. given by P µν t; x, x C t n+ exp where C is a constant. Thus we have the estimate det P µν t; x, x C 3 t nn+ exp where C 3 is a constant. Thus as t 0, therefore r, 3.35 t nr t K t 0, 3.37 Kt K t

30 3.. Heat Determinant Asymptotics From above, we have as t 0, Kt dvolx t n 4πt n M B r x dvolx exp n t σx, x Ht; x, x Now using the coordinate transformation given by.60, the integral over B r x becomes the integral over the Euclidean ball B r/ t ξ. As t 0, r t, therefore we integrate ξ over all R n. Thus Kt t nn+ dvolx dξ g n x exp n 4π n ξ ξ, x Ht; ξ, x. M R n 3.40 Then using the notation for Gaussian average.65, we have Kt t nn+ n π dvolx H, 3.4 n 4π n n n where H n is a function of x. By using 3.4 and 3.5, we see that Ψt; x, x has an asymptotic expansion as t 0 and is given by where, in general, Ψt; x, x M t k ψ k x, x, 3.4 k=0 ψ k x, x = k x, x Px, x a k! k x, x Thus Z, V, W, and S all have asymptotic expansions in non-negative integer powers of t since Ψ can be expanded in non-negative integer powers of t. Therefore we can expand H 3.9 in integer powers of t greater than. Thus we get Ht; x, x x, x k= t k h k x, x, 3.44 where x, x was introduced for convenience. The sum begins from k = since there is a t term in H which can be seen in the S term in the det Y 3.7. Then Kt t nn+ π n 4π n n t k dvolx h k x, x n k= M 5

31 If we define then we have H k = M dvolx h k n, 3.46 Kt t nn+ π n 4π n n t k H k k= Next, we want to calculate h k n. We can expand h k x, x in a covariant Taylor series about the point x. This gives us h k x, x = σ µ σ µ mh k,µ µ x m m=0 where Then we have h k,µ µ m x := m m! h k n = m=0 t m ] µ µm h k x, x ξ µ ξ µ m h n k,µ µ m Evaluating the Gaussian integral and relabeling indices, we obtain h k n = Let us define m=0 t m m! n m m! gµ µ g µ m µ m h k,µ i µ m. 3.5 h k,m x = m! n m m! gµ µ g µ m µ m h k,µ µ m. 3.5 This gives us If we define then h k n = H k,m = M H k = t m h k,m m=0 dvolx h k,m x, 3.54 t m H k,m m=0 6

32 When we plug this all into 3.47, we get Kt t nn+ π n 4π n n k= m=0 When we combine powers of t and re-index the sum, we find t k+m H k,m Kt t nn+ π n 4π n n t k B k, 3.57 k= where B k = k H m,k m m= Note that we can express B k = M dvolx b k x, 3.59 where b k x = k h m,k m x m= It is the coefficients B k that we wish to calculate. In summary, to calculate B k, we need to find b k 3.60 or H k,m 3.54, which are found from h k,m 3.5. To find h k,m, we compute the covariant Taylor expansion coefficients h k,µ µ m 3.49 of h k The coefficients h k are calculated from H 3.9 which is calculated from Y µν 3.9. To compute Y µν, we only need the heat kernel and the Synge function Heat Determinant Asymptotics for Scalar Operators For scalar operators, we do not have a trace in 3. i.e. tr Ψ Ψ = Ψ. Therefore simplify to S µν = σ µ σ ν Ψ, 3.6 V µν = σ µ ΨΨ ν, 3.6 W µν = σ ν ΨΨ µ, 3.63 Z µν = σ µν Ψ + tψψ µν

33 Since we have expanded the heat determinant in powers of t, let us also expand dety thus H in powers of t. First we need to expand S, V, W, and Z in powers of t. Thus from and 3.4, we have S µν =σ µ σ ν ψ0 + σ µσ ν ψ ψ 0 t + σ µ σ ν ψ ψ 0 + ψ t σ µ σ ν ψ 3 ψ 0 + ψ ψ t 3 +, V µν = σ µ ψ 0 ψ 0,ν σ µ ψ,ν ψ 0 + ψ 0,ν ψ t σ µ ψ,ν ψ 0 + ψ,ν ψ t +, 3.66 W µν = σ ν ψ 0 ψ 0,µ σ ν ψ,µ ψ 0 + ψ 0,µ ψ t σ ν ψ,µ ψ 0 + ψ,µ ψ t +, and Z µν = σ µν ψ0 + ψ 0 ψ 0,µν σ µν ψ t + ψ,µν ψ 0 + ψ 0,µν ψ σ µν ψ ψ 0 + ψ t + ψ,µν ψ 0 + ψ,µν ψ + ψ 0,µν ψ σ µν ψ 3 ψ 0 + ψ ψ +, t where each additional index denotes the covariant derivative with respect to x and x respectively. Thus ψ k, µ = µ ψ k, 3.69 and where ψ k, is ψ k with an arbitrary number of indices. ψ k, ν = ν ψ k, 3.70 Expanding H 3.9 in powers of t, we have the first three terms in 3.44: h = { n! E µ µ n E ν ν n n } n σ µ σ ν σ µ ν σ µ n ν n ψn 0, 3.7 h 0 = { n! E µ µ n E ν ν n n σ µ ν σ µ n ν n ψn 0 + n n σ µ ψ 0,ν + σ ν ψ 0,µ σ µ ν σ µ n ν n ψn n n σ µ σ ν σ µ ν σ µ n ν n ψ ψ n 0 8

34 nn + n σ µ σ ν ψ 0,µ ν σ µ ν ψ + nn n σ ν ψ 0,µ σ µ ψ 0,ν σ µ3 ν 3 σ µ n ν n ψn 0 h = n! E µ µ n E ν ν n σ µ3 ν 3 σ µ n ν n ψn 0 }, { n n ψ 0,µ ν σ µ ν ψ σ µ ν σ µ 3 ν 3 ψ 0 + n n σ ν ψ,µ ψ 0 + ψ 0,µ ψ σµ ν σ µ 3 ν 3 ψ 0 + n n σ µ ψ,ν ψ 0 + ψ 0,ν ψ σµ ν σ µ 3 ν 3 ψ 0 + nn n σ µ ψ 0,ν + σ ν ψ 0,µ ψ 0,µ ν σ µ ν ψ σ µ3 ν 3 ψ 0 + n n σ µ σ ν σ µ ν σ µ 3 ν 3 ψ ψ 0 + ψ ψ 0 nn + n σ µ σ ν ψ 0,µ ν σ µ ν ψ σ µ3 ν 3 4ψ ψ 0 nn + n σ µ σ ν ψ,µ ν ψ 0 + ψ 0,µ ν ψ σ µ3 ν 3 ψ 0 nn n σ µ σ ν σ µ ν ψ ψ 0 + ψ σ µ3 ν 3 ψ 0 nn n + n 3 σ µ σ 4 ν ψ 0,µ ν ψ 0,µ3 ν 3 σ µ 3 ν 3 ψ 4ψ 0 nn n n 3 σ µ σ 4 ν σ µ ν ψ ψ 0,µ3 ν 3 σ µ 3 ν 3 ψ 4ψ 0 + nn n σ ν ψ,µ ψ 0 + ψ 0,µ ψ σµ ψ 0,ν σ µ3 ν 3 + nn n σ ν ψ 0,µ σ µ ψ,ν ψ 0 + ψ 0,ν ψ σ µ3 ν 3 + nn n n σ ν ψ 0,µ σ µ ψ 0,ν ψ } 0,µ3 ν 3 σ µ 3 ν 3 ψ σ µ4 ν 4 σ µ n ν n ψn For simplification, we define a matrix F given by F µν = n n! E µα α n E ν β β n σα β σ α n β n Then by using Lemma and letting A µν = σ µν and B µν = γ µν, then F µν = γ µν We note that σ µ σ ν F µν = σ

35 Introducing F µ µ ν ν = n n! E µ µ α α n E ν ν β β n σ α β σ α n β n Then by using Lemma and letting A µν = σ µν and B µν = γ µν, then We note that F µ µ ν ν = γ λ ν γ λ ν δ µ µ λ λ = γ µ ν γ µ ν γ µ ν γ µ ν σ µ σ ν F αµβ ν = σγ αβ σ α σ β, 3.79 and σ µν F αµβ ν = n F αβ = n γ αβ, 3.80 σ µ σ ν σ µ ν Fµ µ ν ν = n σ. 3.8 Lastly, we introduce F µ µ µ 3 ν ν ν 3 = n 3 n 3! E µ µ µ 3 α α n 3 E ν ν ν 3 β β n 3σ α β σ α n 3 β n Then by using Lemma and letting A µν = σ µν and B µν = γ µν, then F µ µ µ 3 ν ν ν 3 = γ λ ν γ λ ν γ λ 3ν 3δ µ µ µ 3 λ λ λ 3 = 3!γ µ ν γ µ ν γ µ 3]ν We note that If we let then σ µν F α α µβ β ν = n F α α β β G α α β β = σµ σ ν F α α µβ β ν, 3.85 G α α β β = σγ α β γ α β + σγ α β γ α β σ β σ α γ α β σ β σ α γ α β + σ β σ α γ α β + σ α σ β γ α β We will now simplify We find that h = σ µσ ν F µν ψ n Hence, using 3.76, h = σψ n

36 Simplifying h 0, we have h 0 =ψ0 n σ µ ψ 0,ν + σ ν ψ 0,µ F µν ψ0 n + σ µ σ ν F µν ψ ψ0 n + σ µ σ ν ψ 0,µ ν σ µ ν ψ F µ µ ν ν ψ0 n + σ ν ψ 0,µ σ µ ψ 0,ν F µ µ ν ν ψ n 0. Plugging in 3.75, 3.76, 3.80, and 3.79, we have Thus, we finally have h 0 =ψ0 n + σ µ ψ 0,ν + σ ν ψ 0,µ γ µν ψ0 n σψ ψ0 n + ψ 0,µν σ µν ψ σγ µν σ µ σ ν ψ0 n ψ 0,µ ψ 0,ν σγ µν σ µ σ ν ψ0 n. h 0 =ψ0 n + σ ν ψ 0,ν + σ µ ψ 0,µ + σψ 0,µν γ µν ψ0 n σ µ σ ν ψ 0,µν + nσψ ψ0 n + σ µ σ ν ψ 0,µ ψ 0,ν σψ 0,µ ψ 0,ν γ µν ψ0 n. Simplifying h, we have h = ψ 0,µν σ µν ψ F µν ψ0 n σ ν ψ,µ ψ 0 + ψ 0,µ ψ + σ µ ψ,ν ψ 0 + ψ 0,ν ψ F µν ψ0 n σ µ ψ 0,ν + σ ν ψ 0,µ ψ 0,µ ν σ µ ν ψ F µ µ ν ν ψ0 n + σ µσ ν F µν ψ ψ 0 + ψ ψ0 n + σ µ σ ν ψ 0,µ ν σ µ ν ψ F µ µ ν ν ψ ψ0 n + σ µ σ ν ψ,µ ν ψ 0 + ψ 0,µ ν ψ F µ µ ν ν ψ0 n σ µ σ ν σ µ ν ψ ψ 0 + ψ F µ µ ν ν ψ0 n + σ µ σ ν ψ 0,µ ν σ µ ν ψ ψ 0,µ3 ν 3 σ µ 3 ν 3 ψ F µ µ µ 3 ν ν ν 3ψ0 n + σ ν ψ,µ ψ 0 + ψ 0,µ ψ σµ ψ 0,ν F µ µ ν ν ψ n σ ν ψ 0,µ σ µ ψ,ν ψ 0 + ψ 0,ν ψ F µ µ ν ν ψ n 3 + σ ν ψ 0,µ σ µ ψ 0,ν ψ 0,µ3 ν 3 σ µ 3 ν 3 ψ 0 F µ µ µ 3 ν ν ν 3ψ n

37 Plugging in 3.75, 3.76, 3.78, 3.79, 3.80, and 3.84, we find { h = σ µν ψ ψ 0,µν γ µν ψ0 + σ ν ψ,µ ψ 0 + ψ 0,µ ψ + σµ ψ,ν ψ 0 + ψ 0,ν ψ γ µν ψ 0 + σ µ ψ 0,ν + σ ν ψ 0,µ γ µν n ψ ψ 0 σ µ ψ 0,ν + σ ν ψ 0,µ ψ 0,µ ν σ ψ ψ 0 + ψ ψ 0 + ψ 0,µν σγ µν σ µ σ ν ψ ψ 0 γ µ ν γ µ ν γ µ ν γ µ ν ψ 0 4 n σψ ψ 0 + ψ,µν ψ 0 + ψ 0,µν ψ σγ µν σ µ σ ν ψ 0 n σ ψ ψ 0 + ψ ψ σ µ σ ν ψ 0,µ ν ψ 0,µ 3 ν 3 Fµ µ µ 3 ν ν ν 3ψ 0 + n ψ 0,µν σγ µν σ µ σ ν ψ ψ 0 n n σψ ψ 0 ψ,µ ψ 0 + ψ 0,µ ψ ψ0,ν σγ µν σ µ σ ν ψ 0,µ ψ,ν ψ 0 + ψ 0,ν ψ σγ µν σ µ σ ν ψ 0,µ ψ 0,ν σγ µν σ µ σ ν n ψ } σ µ σ ν ψ 0,µ ψ 0,ν ψ 0,µ3 ν 3 Fµ µ µ 3 ν ν ν 3 ψ0 n 3. Then simplifying further, we obtain { h = n ψ σψ ψ0 ψ 0,µν γµν ψ0 + ψ,µν σγ µν σ µ σ ν ψ0 + σ µ ψ,µ ψ 0 + ψ 0,µ ψ + σ ν ψ,ν ψ 0 + ψ 0,ν ψ + n σ ν ψ 0,ν + σ µ ψ 0,µ ψ ψ 0 σ ν ψ 0,ν + σ µ ψ 0,µ ψ 0,αβ γ αβ ψ 0 ψ

38 + σ ν ψ 0,β γ µβ + σ µ ψ 0,α γ αν ψ 0,µν ψ 0 n nσψ ψ 0 + n ψ 0,µν σγ µν σ µ σ ν ψ ψ 0 ψ,µ ψ 0 + ψ 0,µ ψ ψ0,ν σγ µν σ µ σ ν ψ 0,µ ψ,ν ψ 0 + ψ 0,ν ψ σγ µν σ µ σ ν n ψ 0,µ ψ 0,ν σγ µν σ µ σ ν ψ } + ψ 0,µ ν ψ 0 ψ 0,µ ψ 0,ν ψ 0,µ ν Gµ µ ν ν ψ0 n Coefficients of Heat Determinant Asymptotic Expansion We want to find the coefficients B k for scalar operators, for which we need the coefficients b k We will be using the coincidence limits derived in Section 5. throughout this section. For our first coefficent, we have From 3.5 and 3.88, and the fact that σ] = 0, h,0 = h ] = b = h, σψ n 0 ] = Thus b = Next, we want to calculate We will first calculate h 0,0. From 3.5, we have b 0 = h, + h 0, h 0,0 = h 0 ] From 3.9, and noting the coincidence limits of σ and its single derivatives are 0, then ] h 0,0 = h 0 ] = = ψ n 0 Next, we will calculate h,. From equation 3.5, h, = n gµ µ µ µ h ]

39 Plugging in h from 3.88, we find h, = n gµ µ µ µ σψ n 0 ]. 3.0 We will only have non-zero terms when both derivatives are on σ, thus since σµν ] = gµν, we have h, = Therefore, we have b 0 = Next, we want to calculate First, we will look at h,0. From 3.5, b = h, + h 0, + h, h,0 = h ] Since coincidence limits of σ and its first derivatives are 0, from 3.94, we are left with only h,0 = n ψ ] + ψ 0, µ µ ] Next, looking at h 0,, from 3.5, h 0, = n gµ µ µ µ h 0 ] Thus from 3.9, h 0, = { n gµ µ µ µ ψ0 n + σβ ψ 0,β ψ0 n + σ α ψ 0,α ψ0 n + σψ 0,αβ γ αβ ψ n 0 σ α σ β ψ 0,αβ ψ n 0 nσψ ψ0 n }]. + σ α σ β ψ 0,α ψ 0,β ψ0 n σψ 0,µ ψ 0,ν γ µν ψ0 n 3.09 For the coincidence limit to not be 0, there must be two derivatives on each σ and no derivatives or two on each ψ 0. Thus h 0, = ν nψ 0, n ν ψ0 n + σ β ν ψ 0,β νψ0 n + σ αν ψ 0,αν ψ0 n + σ ν νψ 0,αβ γ αβ ψ0 n σ αν σ β νψ 0,αβ ψ0 n nσ ν νψ ψ0 n 34 ]. 3.0

40 Now taking the actual coincidence limit, h 0, = + ν ψ 0, n ν ] µ ψ 0, µ ] n ψ ]. 3. Lastly, we need h,. h, = Plugging in h, we find n! gµ µ g µ 3µ 4 µ µ µ3 µ4 h ]. 3. h, = 8n gµ µ g µ 3µ 4 µ µ µ3 µ4 σψ0 n ]. 3.3 The only derivatives that are not zero are the ones with derivatives on σ and on ψ0 n since the symmetrization of 4 indices on σ is 0. Thus we have h, = 4n gµ µ g µ 3µ 4 + g µ µ 3 g µ µ 4 + g µ µ 4 g µ µ 3 Taking the derivatives and coincidence limits, we have σ µ µ µ3 µ4 ψ n 0 ]. 3.4 h, = n + n ψ 0, ν ν ]. 3.5 Therefore, we find that b = n ψ ] + ] µ ψ 0, µ ν ψ 0, 6n ν ]. 3.6 We have calculated b, b 0, and b. Plugging in the coincidence limits from Section 5., we have b = 0, 3.7 b 0 =, 3.8 b = n n + 0 R nq n Therefore, from 3.59, we obtain the first 3 global invariants B = 0, 3.0 B 0 = volm, 3. B = dvol n n + 0 R nq. 3. 7n M 35

41 CHAPTER 4 CONCLUSION We have studied the asymptotic expansion of a new heat invariant, the heat determinant. The heat determinant seems like the next logical step in the study of heat invariants on a manifold. We have calculated the first three coefficients in the asymptotic t-expansion, thus we have Kt t nn+ n π { } B t + B 0 + B t π n n We note that the first term is zero, the second term depends on the volume of the manifold, and the the third term depends on the dimension and the integral of the curvature over the entire manifold. For future work, obviously, we can calculate more coefficients in our expansion, however they do get much more complicated rather quickly. The most important work to be done is to see what this new invariant can tell us about these manifolds, whether we can distinguish between any two manifolds, or if not, how much information about the manifold we can extract from the asymptotic expansion. One could also do more work with the modified heat determinant.03, more specifically, looking at manifolds with a boundary and seeing what kind of information we can extract from this other invariant. One could also study the diagonal of the heat kernel given by K diag t := M dx detp µν t, x, x. 4. This invariant contains much less information than the actual heat determinant, however, it is much easier to calculate higher t terms in our asymptotic expansion. The heat determinant is a first step in the study of non-spectral heat invariants. There is still a lot of work to be done in this area. 36

42 CHAPTER 5 APPENDIX 5. Gaussian Integrals For a real positive symmetric matrix A = A ij, then for any vector B = B i, we have R n dx exp x, Ax + B, x = π n deta exp B, A B, 5. 4 where A = A ij is the inverse of A. Now we shall differentiate the above equation with respect to B i on both sides, for which we obtain R n dx exp x, Ax + B, x x i = π n deta A ik B k exp B, A B. 4 If we differentiate again with respect to B j, we obtain R n dx exp x, Ax + B, x x i x j = π n deta A ij + Aik B k A jl B l exp B, A B If we continue this process and set B = 0, we get the coefficients to the Taylor expansion of 5. in B i and we find that R n dx exp x, Ax x i x ik+ = and R n dx exp x, Ax x i x ik = π n deta k! k k! Ai i A i k i k

43 We define the Gaussian average as f x = π n deta R n dx exp x, Ax f x. 5.6 Thus we have x i x i k+ = 0, 5.7 and x i x i k = k! k k! Ai i A i k i k Evaluation of Coincidence Limits The Synge function is defined in section..7. For more on the Synge function, see 6]. Here, we will evaluate the coincidence limits of the Synge function and its derivatives. We have σ = σ µσ µ = σ ν σν. 5.9 From this, and noting that σ] = 0, we find that σµ ] = By differentiating 5.9, we get σ µ = µ σ ασ α = σ αµ σ α, 5. σ µ = ν σ ασ α = σ αµ σ α. 5. Before taking more derivatives, we shall find the coincidence limit of σ µν. From the definition of σ in.49, we have σ = t t Taking a covariant derivative, we have t σ α = t 0 0 dτ g µν dx µ dτ dx ν dτ. 5.3 dx µ dτ g µν dτ dx ν α dτ

44 Evaluating the covariant derivative, and integrating by parts, we have t σ α =t 0 dτ g µν dx µ dτ =tg µα dx µ dτ δµ α t t =tu α t 0 α dx ν t 0 d x α dτ dτ dτ + Γν γα dτ g µν δ µ α dx γ Γν γα dτ dx γ dτ d x ν dτ dx γ Γν γα dτ dx ν. dτ dx µ dτ 5.5 where U α = dx α dτ. By definition, U α is the vector tangent to the geodesic connecting x and x. We note that the second term in the above equation is just the integral of.46 multiplied by g µα. Thus we have σ α = tu α. 5.6 Thus we have from 5., U µ = U α σ α µ. 5.7 When we take the coincidence limit, we want the limit to not depend on direction, therefore σ µ ν] = δ ν µ 5.8 and ] σµν = gµν. 5.9 Now differentiating again 5. and 5., we have σ µν = ν σαµ σ α = σ αµν σ α + σ αµ σ α ν, 5.0 σ µ ν = ν σ αµ σ α = σ αµ νσ α + σ αµ σ α ν. 5. We note that the derivatives with respect to x always commute with derivatives with respect to x. We also note that we can always commute the first two derivatives since σ is a scalar. Taking a derivative with respect to x of 5.0, we have σ µνβ = σ αµνβ σ α + σ αµν σ α β + σ αµβ σ α ν + σ αµ σ α νβ. 5. Taking the coincidence limit, we have σνµβ ] = σβµν ]. 5.3 Since σ is symmetric in the first two indices, we have from 5.3, σµνβ ] =

45 We also note that by commuting the last two indices in σ µνβ, we have Taking another derivative of 5., we have σ µνβ σ µβν = R α µνβσ α. 5.5 σ µνβγ =σ αµνβγ σ α + σ αµνβ σ α γ + σ αµνγ σ α β + σ αµν σ α βγ + σ αµβγ σ α ν + σ αµβ σ α νγ + σ αµγ σ α νβ + σ αµ σ α νβγ. 5.6 Taking the coincidence limit gives us σγµνβ ] + σβµνγ ] + σνµβγ ] = By commutation of derivatives, we have that σ µνβγ σ µνγβ = R α µβγσ αν + R α νβγσ µα. 5.8 Due to the symmetry of the Riemann tensor, we have σµνβγ ] = σµνγβ ] = σνµβγ ]. 5.9 Taking a derivative of 5.5, we have σ µνβγ σ µβνγ = R α µνβγσ α + R α µνβσ αγ Taking this coincidence limit gives us σµνβγ ] σµβνγ ] = Rγµνβ. 5.3 If we switch β and γ, then add this to the above equation, we find σ µνβγ ] σµβνγ ] σµγνβ ] = Rγµνβ + R βµνγ. 5.3 Using 5.7 and 5.9, the above equations simplifies to To calculate the coincidence limits for primed indices, we just use the formula ] σµνβγ = Rµγνβ + R 3 µβνγ f ] α = f α ] + f α ], 5.34 which is true for any function of x and x 6]. More specifically, we have ] ] ] f µν = ν fµ fµν,

46 In summary, we have calculated the coincidence limits of σ and its derivatives to be and σ] = 0, 5.36 σ µ ] = 0, 5.37 σ ν ] = 0, 5.38 σ µν ] = σ µν ] = σ µ ν ] = g µν, 5.39 σ µνα ] = σ µ να] = σ µ ν α] = σ µ ν α ] = 0, 5.40 σ µνβγ ] = Rµγνβ + R 3 µβνγ, σ µνβγ ] = Rµγνβ + R 3 µβνγ, σ µνβ γ ] = 3 Rµγνβ + R µβνγ, 5.4 σ µν β γ ] = Rµνγβ + R 3 µβγν, σ µ ν β γ ] = Rµγνβ + R 3 µβνγ. We will now calculate the coincidence limits of the Van-Fleck determinant and its derivatives. By definition, = g x g x det σ µν. 5.4 Thus Taking one derivative, we have ] = µ = n! n E α α n E β β nσ α β µσ α β σ α n β n Taking the coincidence limit, since there is a third derivative of σ, we have µ ] = Taking another derivative of 5.44, we have µν = n! n E α α n E β β nσ α β µνσ α β σ α n β n n! n E α α n E β β nσ α β µσ α β νσ α 3 β σ 3 α n β. n 4

47 Using 3.75 and 3.78, we have µν = σ αβ µνγ αβ + σ α β µσ α β ν γ α β γ α β γ α β γ α β Taking the coincidence limit and using 5.4 gives us µν ] = σ α α µν] = 3 R µν Next, we calculate the coincidence limits of ψ k and its derivatives. From 3.43, ψ k = k ak, 5.49 k! where a k can be calculated using the process given in ]. We will only need the first couple coincidence limits which were originally calculated by 7], 5], and 9] and are given by for our operator L = + Q. Thus we have a 0 ] =, 5.50 a ] =Q R, ψ 0 =, 5.53 and from 5.43, we have Taking a single derivative yields ψ 0 ] = ψ 0,µ = µ, 5.55 then using 5.45, we have ψ0,µ ] = Taking one more derivative of ψ 0, we end up with Therefore, using 5.48, we have ψ 0,µν = µν 4 µ ν ψ0,µν ] = 6 R µν

48 To find the coincidence limits of the primed derivatives, we us the formula Next, ψ = a Thus ψ ] = a ] = R Q In summary, we have the coincidence limits ] =, 5.6 µ ] = ν ] = 0, 5.6 µν ] = µ ν ] = 3 R µν, 5.63 µν ] = 3 R µν, 5.64 ψ 0 ] =, 5.65 ψ 0,µ ] =ψ 0,ν ] = 0, 5.66 ψ 0,µν ] =ψ 0,µ ν ] = 6 R µν, 5.67 ψ 0,µν ] = 6 R µν, 5.68 ψ ] = R Q

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