By Virgil Snyder, Charles Herschel Sisam
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We proceed with the construction of c G El\m(E,F) in (3). Let ftp : P —> X be the principal 0(m)-bundle of orthonormal frames of the normal bundle N —> X for the embedding / : X —•> Y, where diraX = n and d i m F = n'. m —> Nx, where m = n' — n and Nx is the fiber of the normal bundle at x G X. In terms of associated bundles, we have N = P x 0 ( m ) Mm = ( P x K m ) / O(m), where O(m) a c t s o n P x R " 1 via (p,v)-A := ( p o i , A~lv). Note that 0(m) also acts on K m + 1 = E m x M via A • (v,a) = (A(v),a), and the m-sphere Sm C E m + 1 is invariant under this action with two fixed points, the poles (0,±1) eSm.
J. Park and K. P. Wojciechowski, Scattering theory and adiabatic decomposition of the (^-determinant of the Dirac Laplacian, Math. Res. Lett. 9 (2002), no. 1, 17-25. 27. J. Park and K. P. Wojciechowski, Adiabatic decomposition of the ^-determinant and Scattering theory, Michigan Math. DG/0111046 . 28. J. Park and K. P. Wojciechowski, Adiabatic decomposition of the zetadeterminant and the Dirichlet to Neumann operator, J. Geom. Phys. 55 (2005), 241-266. 29. J. Park and K. P. Wojciechowski, Agranovich-Dynin formula for the zetadeterminants of the Neumann and Dirichlet problems, Spectral geometry of manifolds with boundary and decomposition of manifolds, 109-121, Contemp.
167-305. 34. K. P. Wojciechowski, The additivity of the rj-invariant. The case of a singular tangential operator, Comm. Math. Phys. 169 (1995), 315-327. 35. K. P. Wojciechowski, The ^-determinant and the additivity of the rj-invariant on the smooth, self-adjoint Grassmannian, Comm. Math. Phys. 201, no. 2 (1999), 423-444. Received by the editors September 15, 2005 ; Revised January 4, 2006 Part II Topological Theories This page is intentionally left blank Analysis, Geometry and Topology of Elliptic Operators, pp.
Analytic Geometry of Space by Virgil Snyder, Charles Herschel Sisam