×

zbMATH — the first resource for mathematics

Théorèmes de l’indice sur les variétés non-compactes. (Index theorems for noncompact manifolds). (French) Zbl 1014.58012
Author’s abstract: We give a formula for the extended index of a Dirac type operator on a complete non-compact Riemannian manifold. For this, we show that this index is the index of the restriction to a compact set together with a boundary condition. This condition comes from a natural elliptic complex on the boundary of the compact set. This result generalizes a result of Atiyah-Patodi-Singer about manifolds with cylindrical end.
Reviewer: W.Mozgawa (Lublin)

MSC:
58J20 Index theory and related fixed-point theorems on manifolds
58J32 Boundary value problems on manifolds
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ackermann J, Math. 471 pp 23– (1996)
[2] Houston J. Math. 19 (2) pp 223– (1993)
[3] J. Funct. Anal. 119 pp 19– (1994)
[4] Atiyah V. K., Math. Proc. Camb. Phil. Soc. 77 pp 43– (1975)
[5] [A-S-S] J. Avron, R. Seiler, B. Simon, The index of a pair of projections, J. Funct. Anal. 120, n2 (1994), 220-237. · Zbl 0822.47033
[6] Bo B, Tbilis pp 45– (1979)
[7] [B-W] B. Booss, K. Wojciechowski, Elliptic boundary problems for Dirac Operators, BirkhaEuser, Boston 1993. · Zbl 0836.58041
[8] Borisov W., Comm. Math. Phys. 114 pp 475– (1988)
[9] Br J., Geom. 32 pp 491– (1990)
[10] Br J., Amer. J. Math. 110 pp 659– (1988)
[11] Bu U., J. Funct. Anal. 105 pp 63– (1992)
[12] [Ca] A. P. CalderoAn, Boundary value problems for ellipic equations, Outlines of the Joint Soviet-American Symposium on Partial Di erential Equations, Novosibirsk 1963.
[13] Carron, Math. 198 pp 81– (2001)
[14] [C2] G. Carron, L2-harmonicforms on some asymptotically at manifold, PreApublication, 2000.
[15] Ch A. W, Trans. Amer. Math. Soc. 289 pp 1– (1985)
[16] [Da] E. B. Davies, Spectral theory and di erential operators, Cambridge stud. adv. Math. (1995).
[17] Dodziuk, Ann. Math. Stud. 102 pp 291– (1982)
[18] Do H, J. Funct. Anal. 75 pp 362– (1987)
[19] [G-L] M. Gromov, H. B. Lawson, Jr., Positive scalar curvature and the Dirac operator on a complete Riemannian manifold, Publ. Math. I.H.E.S. 58 (1983), 83-196.
[20] [H] L. HoErmander, The analysis of linear partial di erential operator, Vol. III, Springer-Verlag, New-York 1985.
[21] [P] R. Palais, Seminar on the Atiyah-Singer index theorem, Ann. Math. Stud. (1965). · Zbl 0137.17002
[22] [R1] J. Roe, A note on the relative index theorem, Quart. J. Mat. Oxford (2) 42 (1991), 365-373. · Zbl 0746.19009
[23] [R2] J. Roe, Partitioning non-compact manifolds and the dual Toeplitz problem, in: Operator Algebras and Applications, Cambridge University Press (1989), 187-228.
[24] Amer. J. Math. 88 pp 781– (1966)
[25] Taylor II, Appl. Math. Sci. pp 116– (1996)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.