LEADER 04114nam 2200637Ia 450 001 9910454372203321 005 20200520144314.0 010 $a1-281-95164-1 010 $a9786611951641 010 $a981-281-010-2 035 $a(CKB)1000000000538027 035 $a(EBL)1679404 035 $a(SSID)ssj0000178208 035 $a(PQKBManifestationID)11156230 035 $a(PQKBTitleCode)TC0000178208 035 $a(PQKBWorkID)10221344 035 $a(PQKB)10244395 035 $a(MiAaPQ)EBC1679404 035 $a(WSP)00004691 035 $a(Au-PeEL)EBL1679404 035 $a(CaPaEBR)ebr10255391 035 $a(CaONFJC)MIL195164 035 $a(OCoLC)815754701 035 $a(EXLCZ)991000000000538027 100 $a20010307d2001 uy 0 101 0 $aeng 135 $aur|n|---||||| 181 $ctxt 182 $cc 183 $acr 200 14$aThe index theorem and the heat equation method$b[electronic resource] /$fYanlin Yu 210 $aSingapore ;$aRiver Edge, NJ $cWorld Scientific$dc2001 215 $a1 online resource (309 p.) 225 1 $aNankai tracts in mathematics ;$vv. 2 300 $aDescription based upon print version of record. 311 $a981-02-4610-2 320 $aIncludes bibliographical references (p. 279-282) and index. 327 $aPREFACE; CONTENTS; DEFINITIONS AND FORMULAS; CHAPTER 1 PRELIMINARIES IN RIEMANNIAN GEOMETRY; 1.1 Basic Notions of Riemannian Geometry; 1.2 Computations by using Orthonormal Moving Frame; 1.3 Differential Forms and Orthonormal Moving Frame Method; 1.4 Classical Geometric Operators; 1.5 Normal Coordinates; 1.6 Computations on Sphere; 1.7 Connections on Vector Bundles and Principal Bundles; 1.8 General Tensor Calculus; CHAPTER 2 SCHRODINGER AND HEAT OPERATORS; 2.1 Fundamental Solution and Levi Iteration; 2.2 Existence of Fundamental Solution; 2.3 Cauchy Problem of Heat Equation 327 $a2.4 Hodge Theorem2.5 Applications of Hodge Theorem; 2.6 Index Problem; CHAPTER 3 MP PARAMETRIX AND APPLICATIONS; 3.1 MP Parametrix; 3.2 Existence of Initial Solutions; 3.3 Asymptotic Expansion for Heat Kernel; 3.4 Local Index for Elliptic Operators; CHAPTER 4 CHERN-WEIL THEORY; 4.1 Characteristic Forms and Characteristic Classes; 4.2 General Characteristic Forms; 4.3 Chern Root Algorithm; 4.4 Formal Approach to Local Index of Signature Operator; CHAPTER 5 CLIFFORD ALGEBRA AND SUPER ALGEBRA; 5.1 Clifford Algebra; 5.2 Super Algebra; 5.3 Computations on Supertraces; CHAPTER 6 DIRAC OPERATOR 327 $a6.1 Spin Structure6.2 Spinor Bundle; 6.3 Dirac Operator; 6.4 Index of Dirac Operator; CHAPTER 7 LOCAL INDEX THEOREMS; 7.1 Local Index Theorem for Dirac Operator; 7.2 Local Index Theorem for Signature Operator; 7.3 Local Index Theorem for de Rham-Hodge Operator; CHAPTER 8 RIEMANN-ROCH THEOREM; 8.1 Hermitian Metric; 8.2 Hermitian Connection; 8.3 Riemann-Roch Operator; 8.4 Weitzenbock Formula; 8.5 Index Theorem; 8.6 Riemann-Roch Operator in Complex Analysis; REFERENCES; INDEX 330 $aThis book provides a self-contained representation of the local version of the Atiyah-Singer index theorem. It contains proofs of the Hodge theorem, the local index theorems for the Dirac operator and some first order geometric elliptic operators by using the heat equation method. The proofs are up to the standard of pure mathematics. In addition, a Chern root algorithm is introduced for proving the local index theorems, and it seems to be as efficient as other methods. Contents: Preliminaries in Riemannian Geometry; Schro?dinger and Heat Operators; MP Parametrix and Applications; Chern-Weil Th 410 0$aNankai tracts in mathematics ;$vv. 2. 606 $aAtiyah-Singer index theorem 606 $aHeat equation 608 $aElectronic books. 615 0$aAtiyah-Singer index theorem. 615 0$aHeat equation. 676 $a514.74 700 $aYu$b Yanlin$0880508 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910454372203321 996 $aThe index theorem and the heat equation method$91966077 997 $aUNINA