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Families index for manifolds with boundary, superconnections, and cones. I: Families of manifolds with boundary and Dirac operators. (English) Zbl 0696.53021

This is the first part of a work of the authors in order to establish a formula for the Chern character of a family of Dirac operators of Atiyah- Patodi-Singer on even dimensional manifolds with boundary. First, they establish the index theorem of Atiyah, Patodi and Singer for one single manifold with boundary, by using the cone method of the second author [see Proc. Natl. Acad. Sci. USA 76, 2103-2106 (1976; Zbl 0411.58003)]. Then they construct the Levi-Civita superconnection on a family of manifolds with isolated conical singularities and prove the existence of both the index bundle associated with the family of Dirac operators in the sense of M. F. Atiyah, V. K. Patodi and I. M. Singer [Math. Proc. Camb. Philos. Soc. 78, 405-432 (1975; Zbl 0314.58016)] and the index bundle associated with the family of Dirac operators in the sense of Cheeger.
Reviewer: A.Bejancu

MSC:

53C05 Connections (general theory)
58J20 Index theory and related fixed-point theorems on manifolds
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References:

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