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Selected Topics in Geometric Analysis Selected Topics in Geometric Analysis by Chow Ha Tak Thesis Submitted to the Faculty of the Graduate School of The Chinese University of Hong Kong (Division of Mathematics) In partial fulfillment of the requirements for the Degree of Master of Philosophy June, 1998 p/^( ]jj^jt]B¾ %^4J ^<^^0X 摘要 在本論文中，我們會探討 Laplacian 算子的一些性質和熱方程方法的 應用。在文中，我們只考慮緊致、可定向、連通無邊的黎曼流形。 設 (M" ， g) 為 n 維流形 'g 為 M 上的一個黎曼度量。我們會在第一章 中給出一些基本的定義及引入一些有用的符號。在第二章，我們會 應用熱方程方法去証明 Hodge 定理。 Hodge 定理指出作用於 COOA"T. M 的 Laplacian 算子的特徵值 λ 可以排列起來，而且有 COOAPT. M 的一個么正基。這使我們得到 Hodge 分解及有 Ker /1"與 H，~?( M) 為同構。由此我們得出 Euler 示性數跟熱算子 e -/ð.P 的熱核 叭叭 ， 吋)的關係x χ以(M)恃至 乏(卜一 l)P吋叫>}d副imH的;;本μj品以}μ(M)= 乏(←一 l)P吋叫7dωimKe叮叫rLXρ =2:(←一 l )fJTr e- /6P = 2: (←一 1) 1' fM tr 叭圳e PJ(t川， 同時由熱核的漸近展開式及 χ以(M的)與時間無關，我們可得 χ以(川川M的)=(μ4何州π )-吋→叫叫川川叮，;川/甩11 127j Bonnet 定理指出 (4π )-1112 L二。(一 1)" tr uf (x, x )dvol 為 Euler 式及可以用曲 率張量表示。在証明 Chem-Gauss-Bonnert 定理中，我們會使用超跡 及 Weitzenböck 公式。超跡的計算及 Weitzenböck 公式的証明會在第 三章出現。而 Chem-Gauss-Bonnert 定理的証明會在第五章。我們會 在最後一節介紹一下 Atiyah-Singer 指標定理。詳細及深入的討論可 以參考 [12] 、 [13] 、 [15] 及 [17] 。 Seclected topic in geometric analysis 1 Abstract In this thesis, we will study the properties of the Laplacian and the heat equa- tion method. We will only consider a Riemannian manifold (M", g) which is com- pact, oriented, connected and without boundary. In chapter 1，we give the basic definitions and introduce some notations. In chapter 2, we use the heat equation approach to prove the Hodge theorem, which states that there exists an orthonor- mal basis of C°°A^T*M consisting of eigen-vectors of the Laplacian, leading to the 〜 Hodge decomposition and the isomorphism h : Ker bP ^ H^j^{M) by u) ^~"> [u • From this, we have the relation between the Euler characteristic and the kernel of AP, x{M) = ^{-l)PdimH^j^{M) = X)(-l)MimKerA^'. On the other hand, we have X;(-l)MimKerA 二 X)(-l)^Tre"^^" 二 E(_lF /Mtre^(^,x,x), where e^{t, X, y) is the heat kernel of the heat operator e_," acting on smooth p-forms. At the same time, e^(t, x, x) has the asymptoic expansion (47rt)"^/^ Y^T=o 以【(工,工)一. Finally, since x(M) is independent of t, we get X(M) = (47r)-^/2 [ ^{-l)Ptiul{x,x) dvol )M p=o where k = n/2. The Chern-Gauss-Bonnet theorem says that n (47T)-"/2 ^{-l)Ptlul{x, X) dV0l p=0 is the Euler form and can be expresses by the components Rijki of the curvature tensor. In the proof of the Chern-Gauss-Bonnet theorem, we use the supertrace and the Weitzenbock formula which will be discussed in the chapter 3. The proof will be given in chapter 5. We construct the heat kernel in chapter 4. The basic work will be done in chapter 3. In the final section, we give an introduction of the Atiyah-Singer Index theorem, drawing on the detailed discussions in [12], [13], 15] and [17 . Seclected topic in geometric analysis 2 ACKNOWLEDGMENTS I wish to express my gratitude to my supervisor Prof. Luk Hing Sun. He gave me much advice and help. In particular, he does not mind having a student like me and spends his time to teach me. Thank you very much! I would like to acknowledge the help of Prof. Steven Rosenberg from Boston University. A special thank is due to Prof. Tom Parker for sending his unpublished article [12 to me. And then, I would like to ackonwledge the help of my classmate, Tang wun-cheung, Yuen wai-ching, Keung yee-lo. At the end, I want to say thanks to Keith Law. He gave me many technical support about the software, LATEX. Thank you. Contents 1 The Laplacian on a Riemannian Manifold 5 1.1 Riemannian metrics 5 1.2 L^ Spaces of Functions and Forms 6 1.3 The Laplacian on Functions and Forms 8 2 Hodge Theory for Functions and Forms 14 2.1 Analytic Preliminaries 14 2.2 The Hodge Theorem for Functions 20 2.3 The Hodge Theorem for Forms 27 2.4 Regularity Results 29 2.5 The Kernel of the Laplacian on Forms 33 3 Fermion Calculus and Weitzenb6ck Formula 36 3.1 The Levi-Civita Connection 36 3.2 Fermion calculus 39 3.3 Weitzenb6ck Formula, Bochner Formula and Garding's Inequality 53 3.4 The Laplacian in Exponential Coordinates 59 3 Seclected topic in geometric analysis 4 4 The Construction of the Heat Kernel 63 4.1 Preliminary Results for the Heat Kernel 63 4.2 Construction of the Heat Kernel 66 4.2.1 Construction of the Parametrix 66 4.2.2 The Heat Kernel for Functions 70 4.2.3 The Heat Kernel for Forms 76 4.3 The Asymptotics of the Heat Kernel 77 5 The Heat Equation Approach to the Chern-Gauss- Bonnet Theorem 82 5.1 The Heat Equation Approach 82 5.2 Proof of the Chern-Gauss-Bonnet Theorem 85 5.3 Introduction to Atiyah-Singer Index Theorem . 87 5.3.1 A Survey of Characteristic Forms 87 5.3.2 The Hirzenbruch Signature Theorem 90 5.3.3 The Atiyah-Singer Index Theorem 93 Bibliography Notation index Chapter 1 The Laplacian on a Riemannian Manifold In this chapter we present the basic definitions and facts to be used in the sub- sequent chapters. 1.1 Riemannian metrics Let M be an n-dimensional, orientable, connected, compact Riemannian manifold without boundary. For each point x e M, the tangent space to M at x will be denoted by T^M and the tangent bundle of M will be denoted by TM. The cotangent space and the cotangent bundle will be denoted by T*M and T*M respectively. The Riemannian metric g on M associates to each x G M an inner product on T^M, which we denote by gJ^�•). Note that since Qx is a bilinear form on T^M, it is an element of T*M^T*M. To say that g varies smoothly just means that g is a smooth section of the bundle T*M^T*M. We express the Riemannian metric in local coordinates as follows. Let (xi,...，Xn) be coordinates near x and di denotes 嘉.Sometime, we also denote 嘉 by ^. 5 Seclected topic in geometric analysis 6 Any V, w G T^M can be written as n n V = ^ ai di, w = ^ A di. i=l i=l We have n n ffx(v, U)) = 9x{^ Oiidi, ^ 0jdj) i=l j=l =^c^iPj9x{di, dj). i,3 Thus, Qx is determined by the symmetric, positive definite matrix {oiA^)) = [gx{di, dj)^ Note that while the metric g is defined on all of M, the gij{x) are defined only in a coordinate chart. We write ^ ^ g 二 2_^ 9ij dx^ 0 dx] i,j where dx^ is dual to di and gij are smooth functions for all i, j. Let hij — 9x(dy.,dyj) be the matrix of the metric g in another coordinate chart with coor- dinates (yi,...，yn), then on the overlap of the charts, we have ^ dxk dxi hf\^‘W，i. We also have dy^ = {dyi/dxk)dx^ and Y^^,i 9ki dx^ 0 dx^ 二 J2i,j hij dy' (g) dyK 1.2 Z/2 Spaces of Functions and Forms Definition 1.2.1 The volume form of a Riemannian metric g is defined to be the top dimensional form dvol which in local coordinates is given by dvol 二 ^det^ dx^ 八...A dx^, Seclected topic in geometric analysis 7 whenever (¾!,...,氏„) is a positively oriented basis ofT^M and detg = det{gij). The volume of (M, g) is defined to be vol{M) = / dvol{x). JM Let (x/i, ...，yn) are another coordinates at x and h = {gx{dyi,〜)）then Vdeth^iA...A^n = ^det(^g:,{dy, , ^.)) det^|^) dx^ A ... A dx^ =det (^) det (^) y/^dx' A ... A dx^ =\Jdet g dx^ 八..•八 cb:\ Definition 1.2.2 The Hilbert space L^{M, g) is defined to be the completion of C^{M) (the set of smooth functions on M with compact support) with respect to the inner product (/, g) = / f{x)g{x) dvoL JM Let C^N^T*M be the space of smooth differential forms of degree p on M, p 二 0，1,... ,71. C^h*T*M denotes the space of smooth differential form of mixed degree. Recall that if V is a finite dimensional vector space with an inner product 〈.，•〉，then there is an isomorphism a : V ~^ F* where a{v) = v* satisfies v*{w) — {v, w) for v, w G V. V* inherits an inner product, also denoted〈•，.〉，given by {v*, w*) = {v, w). In particular, a Riemannian metric g on a manifold induces an inner prod- uct, also denoted g, on each cotangent space T*M under the isomorphism a constructed above. We also use a to denote the bundle isomorphism a : TM ^^ T*M induced by a in each fiber. In local coordinates, set g” 二 g(doci,docj) then (p") = {9ij)'^- Seclected topic in geometric analysis 8 Definition 1.2.3 The Hilhert space L^A^T*M is defined to be the completion of C^A^T*M with respect to the global inner product {^,ri) = / p(o;,7/) dvol{x). JM Similarly, the inner product g induces an inner product g on T^M®.. .<^TxM, hence on each hPT*M.
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