LEADER 04790nam 2200673Ia 450 001 9910456151603321 005 20200520144314.0 010 $a981-238-658-0 035 $a(CKB)111087028338018 035 $a(EBL)1679402 035 $a(OCoLC)879023527 035 $a(SSID)ssj0000190857 035 $a(PQKBManifestationID)11215790 035 $a(PQKBTitleCode)TC0000190857 035 $a(PQKBWorkID)10181009 035 $a(PQKB)11528095 035 $a(MiAaPQ)EBC1679402 035 $a(WSP)00004756 035 $a(PPN)164252266 035 $a(Au-PeEL)EBL1679402 035 $a(CaPaEBR)ebr10255555 035 $a(CaONFJC)MIL530336 035 $a(EXLCZ)99111087028338018 100 $a20010823d2001 uy 0 101 0 $aeng 135 $aur|n|---||||| 181 $ctxt 182 $cc 183 $acr 200 10$aLectures on Chern-Weil theory and Witten deformations$b[electronic resource] /$fWeiping Zhang 210 $aRiver Edge, N.J. $cWorld Scientific$dc2001 215 $a1 online resource (131 p.) 225 1 $aNankai tracts in mathematics ;$v4 300 $aDescription based upon print version of record. 311 $a981-02-4685-4 320 $aIncludes bibliographical references and index. 327 $aContents; Preface; Chapter 1 Chern-Weil Theory for Characteristic Classes; 1.1 Review of the de Rham Cohomology Theory; 1.2 Connections on Vector Bundles; 1.3 The Curvature of a Connection; 1.4 Chern-Weil Theorem; 1.5 Characteristic Forms, Classes and Numbers; 1.6 Some Examples; 1.6.1 Chern Forms and Classes; 1.6.2 Pontrjagin Classes for Real Vector Bundles; 1.6.3 Hirzebruch's L-class and A-class; 1.6.4 K-groups and the Chern Character; 1.6.5 The Chern-Simons Transgressed Form; 1.7 Bott Vanishing Theorem for Foliations; 1.7.1 Foliations and the Bott Vanishing Theorem 327 $a1.7.2 Adiabatic Limit and the Bott Connection1.8 Chern-Weil Theory in Odd Dimension; 1.9 References; Chapter 2 Bott and Duistermaat-Heckman Formulas; 2.1 Berline-Vergne Localization Formula; 2.2 Bott Residue Formula; 2.3 Duistermaat-Heckman Formula; 2.4 Bott's Original Idea; 2.5 References; Chapter 3 Gauss-Bonnet-Chern Theorem; 3.1 A Toy Model and the Berezin Integral; 3.2 Mathai-Quillen's Thom Form; 3.3 A Transgression Formula; 3.4 Proof of the Gauss-Bonnet-Chern Theorem; 3.5 Some Remarks; 3.6 Chern's Original Proof; 3.7 References; Chapter 4 Poincare-Hopf Index Formula: an Analytic Proof 327 $a4.1 Review of Hodge Theorem4.2 Poincare-Hopf Index Formula; 4.3 Clifford Actions and the Witten Deformation; 4.4 An Estimate Outside of Up zero(V) Up; 4.5 Harmonic Oscillators on Euclidean Spaces; 4.6 A Proof of the Poincare-Hopf Index Formula; 4.7 Some Estimates for DT,i's, 2 i 4; 4.8 An Alternate Analytic Proof; 4.9 References; Chapter 5 Morse Inequalities: an Analytic Proof; 5.1 Review of Morse Inequalities; 5.2 Witten Deformation; 5.3 Hodge Theorem for ( * (M), dTf; 5.4 Behaviour of rf Near the Critical Points of f; 5.5 Proof of Morse Inequalities; 5.6 Proof of Proposition 5.5 327 $a5.7 Some Remarks and Comments5.8 References; Chapter 6 Thom-Smale and Witten Complexes; 6.1 The Thom-Smale Complex; 6.2 The de Rham Map for Thom-Smale Complexes; 6.3 Witten's Instanton Complex and the Map eT; 6.4 The Map P, TeT; 6.5 An Analytic Proof of Theorem 6.4; 6.6 References; Chapter 7 Atiyah Theorem on Kervaire Semi-characteristic; 7.1 Kervaire Semi-characteristic; 7.2 Atiyah's Original Proof; 7.3 A proof via Witten Deformation; 7.4 A Generic Counting Formula for k(M ); 7.5 Non-multiplicativity of k(M); 7.6 References; Index 330 $a This invaluable book is based on the notes of a graduate course on differential geometry which the author gave at the Nankai Institute of Mathematics. It consists of two parts: the first part contains an introduction to the geometric theory of characteristic classes due to Shiing-shen Chern and Andre? Weil, as well as a proof of the Gauss-Bonnet-Chern theorem based on the Mathai-Quillen construction of Thom forms; the second part presents analytic proofs of the Poincare?-Hopf index formula, as well as the Morse inequalities based on deformations introduced by Edward Witten.
Contents: 410 0$aNankai tracts in mathematics ;$vv. 4. 606 $aChern classes 606 $aIndex theorems 606 $aComplexes 608 $aElectronic books. 615 0$aChern classes. 615 0$aIndex theorems. 615 0$aComplexes. 676 $a514.72 676 $a516.36 700 $aZhang$b Weiping$0924813 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910456151603321 996 $aLectures on Chern-Weil theory and Witten deformations$92075783 997 $aUNINA