Jean-Michel Bismut is professor of mathematics at the University of Paris-Sud. Gilles Lebeau is professor of mathematics at the University of Nice Sophia-Antipolis.

• *Frontmatter, pg. i*Contents, pg. v*Introduction, pg. 1*Chapter 1. Elliptic Riemann-Roch-Grothendieck and flat vector bundles, pg. 11*Chapter 2. The hypoelliptic Laplacian on the cotangent bundle, pg. 25*Chapter 3. Hodge theory, the hypoelliptic Laplacian and its heat kernel, pg. 44*Chapter 4. Hypoelliptic Laplacians and odd Chern forms, pg. 62*Chapter 5. The limit as t --> + and b --> 0 of the superconnection forms, pg. 98*Chapter 6. Hypoelliptic torsion and the hypoelliptic Ray-Singer metrics, pg. 113*Chapter 7. The hypoelliptic torsion forms of a vector bundle, pg. 131*Chapter 8. Hypoelliptic and elliptic torsions: a comparison formula, pg. 162*Chapter 9. A comparison formula for the Ray-Singer metrics, pg. 171*Chapter 10. The harmonic forms for b --> 0 and the formal Hodge theorem, pg. 173*Chapter 11. A proof of equation (8.4.6), pg. 182*Chapter 12. A proof of equation (8.4.8), pg. 190*Chapter 13. A proof of equation (8.4.7), pg. 194*Chapter 14. The integration by parts formula, pg. 214*Chapter 15. The hypoelliptic estimates, pg. 224*Chapter 16. Harmonic oscillator and the J0 function, pg. 247*Chapter 17. The limit of A'2phib,+-H as b --> 0, pg. 264*Bibliography, pg. 353*Subject Index, pg. 359*Index of Notation, pg. 361

Jean-Michel Bismut is professor of mathematics at the University of Paris-Sud. Gilles Lebeau is professor of mathematics at the University of Nice Sophia-Antipolis.

• *Frontmatter, pg. i*Contents, pg. v*Introduction, pg. 1*Chapter 1. Elliptic Riemann-Roch-Grothendieck and flat vector bundles, pg. 11*Chapter 2. The hypoelliptic Laplacian on the cotangent bundle, pg. 25*Chapter 3. Hodge theory, the hypoelliptic Laplacian and its heat kernel, pg. 44*Chapter 4. Hypoelliptic Laplacians and odd Chern forms, pg. 62*Chapter 5. The limit as t --> + and b --> 0 of the superconnection forms, pg. 98*Chapter 6. Hypoelliptic torsion and the hypoelliptic Ray-Singer metrics, pg. 113*Chapter 7. The hypoelliptic torsion forms of a vector bundle, pg. 131*Chapter 8. Hypoelliptic and elliptic torsions: a comparison formula, pg. 162*Chapter 9. A comparison formula for the Ray-Singer metrics, pg. 171*Chapter 10. The harmonic forms for b --> 0 and the formal Hodge theorem, pg. 173*Chapter 11. A proof of equation (8.4.6), pg. 182*Chapter 12. A proof of equation (8.4.8), pg. 190*Chapter 13. A proof of equation (8.4.7), pg. 194*Chapter 14. The integration by parts formula, pg. 214*Chapter 15. The hypoelliptic estimates, pg. 224*Chapter 16. Harmonic oscillator and the J0 function, pg. 247*Chapter 17. The limit of A'2phib,+-H as b --> 0, pg. 264*Bibliography, pg. 353*Subject Index, pg. 359*Index of Notation, pg. 361

Jean-Michel Bismut is professor of mathematics at the University of Paris-Sud. Gilles Lebeau is professor of mathematics at the University of Nice Sophia-Antipolis.

• *Frontmatter, pg. i*Contents, pg. v*Introduction, pg. 1*Chapter 1. Elliptic Riemann-Roch-Grothendieck and flat vector bundles, pg. 11*Chapter 2. The hypoelliptic Laplacian on the cotangent bundle, pg. 25*Chapter 3. Hodge theory, the hypoelliptic Laplacian and its heat kernel, pg. 44*Chapter 4. Hypoelliptic Laplacians and odd Chern forms, pg. 62*Chapter 5. The limit as t --> + and b --> 0 of the superconnection forms, pg. 98*Chapter 6. Hypoelliptic torsion and the hypoelliptic Ray-Singer metrics, pg. 113*Chapter 7. The hypoelliptic torsion forms of a vector bundle, pg. 131*Chapter 8. Hypoelliptic and elliptic torsions: a comparison formula, pg. 162*Chapter 9. A comparison formula for the Ray-Singer metrics, pg. 171*Chapter 10. The harmonic forms for b --> 0 and the formal Hodge theorem, pg. 173*Chapter 11. A proof of equation (8.4.6), pg. 182*Chapter 12. A proof of equation (8.4.8), pg. 190*Chapter 13. A proof of equation (8.4.7), pg. 194*Chapter 14. The integration by parts formula, pg. 214*Chapter 15. The hypoelliptic estimates, pg. 224*Chapter 16. Harmonic oscillator and the J0 function, pg. 247*Chapter 17. The limit of A'2phib,+-H as b --> 0, pg. 264*Bibliography, pg. 353*Subject Index, pg. 359*Index of Notation, pg. 361