Spectrum and the fixed point sets of isometries. III
Remark added in 2000. This paper was written in 1976 and has remained unpublislied for twenty five years. For a survey of many subsequent developments, the reader should consult the textbook by \textit{N. Berline}, \textit{E. Getzler}, and \textit{M. Vergne}, entitled ``Heat kernels and Dirac Operators'' (1992; Zbl 0744.58001).NEWLINENEWLINENEWLINEFrom the introduction: Let \(M\) be a compact Riemannian manifold of dimension \(d\) and \(f:M\to M\) an isometry with fixed point set \(\Omega\). \(\Omega\) is the disjoint union of closed connected submanifolds \(N\) of dimension \(n\). If \(\Delta\) denotes the Laplace operator of \(M\) with eigenvalues \(\lambda\), then \(f\) induces linear maps \(f^\#_\lambda\) on the eigenspaces of \(\Delta\). For each \(N\in\Omega\) and \(a\in N\), \(f\) induces an \(0(d-n)\) action \(A:(T_aN)^\perp \to(T_aN)^\perp\) on the fiber of the normal bundle \((TN)^\perp\) to \(N\) at \(a\). \((I-A)\) is invertible and we denote \(B=(I-A)^{-1}\). NEWLINENEWLINENEWLINEIn part I of this paper [\textit{H. Donnelly}, Math. Ann. 224, 161-170 (1976; Zbl 0327.53031)] the following asymptotic expansion was derived: NEWLINE\[NEWLINE\sum_\lambda \text{Tr} (f_\lambda^\#) e^{-t\lambda} \sim\sum_{N\in \Omega}(4 \pi t)^{-{n\over 2}}\sum^\infty_{k=0} t^k\int_N b_k(f,a)d vol_N(a).NEWLINE\]NEWLINE It was also shown that the \(b_k(f,a)\) have a simple description using invariant theory. This suggested a new proof of the Atiyah-Singer-Lefschetz formulas for compact group actions. In Part II [\textit{H. Donnelly} and \textit{V. K. Patodi}, Topology 16, 1-11 (1977; Zbl 0341.53023)] we derived such a proof of the \(G\)-signature theorem of Atiyah-Singer.NEWLINENEWLINENEWLINEIn this paper (Part III) we present the additional details needed to deduce the \(G\)-signature theorem with coefficients in a Hermitian vector bundle. While the methods of Part II clearly generalize, it is nevertheless of some interest to write down these results in a careful manner. In particular one may apply the generalized theorem to deduce similar results for the other classical elliptic complexes.NEWLINENEWLINEFor the entire collection see [Zbl 0971.00008].
scientific article; zbMATH DE number 1670978