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A direct geometric proof of the Lefschetz fixed point formulas. (English) Zbl 0747.58016

It is presented a very simple and direct geometric proof of the Lefschetz fixed point formulas by computing the equivariant index of the Dirac operator with respect to an isometry of the base manifold. The paper may be seen as a completion of the program initiated by V. K. Patodi in J. Differ. Geom. 5, 251–283 (1971; Zbl 0219.53054) and J. Differ. Geom. 5, 233–249 (1971; Zbl 0211.53901), studying the local asymptotics of the heat kernel for the de Rham complex.

MSC:

58C30 Fixed-point theorems on manifolds
53B20 Local Riemannian geometry
57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
53A55 Differential invariants (local theory), geometric objects
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[1] M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes. I, Ann. of Math. (2) 86 (1967), 374 – 407. · Zbl 0161.43201
[2] M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes. II. Applications, Ann. of Math. (2) 88 (1968), 451 – 491. · Zbl 0167.21703
[3] Nicole Berline and Michèle Vergne, A computation of the equivariant index of the Dirac operator, Bull. Soc. Math. France 113 (1985), no. 3, 305 – 345 (English, with French summary). · Zbl 0592.58044
[4] Jean-Michel Bismut, The Atiyah-Singer theorems: a probabilistic approach. I. The index theorem, J. Funct. Anal. 57 (1984), no. 1, 56 – 99. , https://doi.org/10.1016/0022-1236(84)90101-0 Jean-Michel Bismut, The Atiyah-Singer theorems: a probabilistic approach. II. The Lefschetz fixed point formulas, J. Funct. Anal. 57 (1984), no. 3, 329 – 348. · Zbl 0556.58027
[5] Ezra Getzler, A short proof of the local Atiyah-Singer index theorem, Topology 25 (1986), no. 1, 111 – 117. · Zbl 0607.58040
[6] Mark W. Goodman, Proof of character-valued index theorems, Comm. Math. Phys. 107 (1986), no. 3, 391 – 409. · Zbl 0613.58031
[7] V. K. Patodi, An analytic proof of Riemann-Roch-Hirzebruch theorem for Kaehler manifolds, J. Differential Geometry 5 (1971), 251 – 283. · Zbl 0219.53054
[8] V. K. Patodi, Curvature and the eigenforms of the Laplace operator, J. Differential Geometry 5 (1971), 233 – 249. · Zbl 0211.53901
[9] Yan Lin Yu, Local index theorem for Dirac operator, Acta Math. Sinica (N.S.) 3 (1987), no. 2, 152 – 169. · Zbl 0643.58020
[10] Wei Ping Zhang, Local Atiyah-Singer index theorem for families of Dirac operators, Differential geometry and topology (Tianjin, 1986 – 87) Lecture Notes in Math., vol. 1369, Springer, Berlin, 1989, pp. 351 – 366.
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