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Register a new userA Stefan-Sussmann theorem for normal distributions on manifolds with boundary
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An analogue of the Stefan-Sussmann Theorem on manifolds with boundary is proven for normal distributions. These distributions contain vectors transverse to the boundary along its entirety. Plain integral manifolds are not enough to integrate a normal distribution; the next best integrals are so-called neat integral manifolds with boundary. The conditions on the distribution for this integrability is expressed in terms of adapted collars and integrability of a pulled-back distribution on the interior and on the boundary.
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We study the index of the APS boundary value problem for a strongly Callias-type operator $D$ on a complete even dimensional Riemannian manifold $M$ (the odd dimensional case was considered in our previous paper arXiv:1706.06737). We use this index to define the relative $eta$-invariant $eta(A_1,A_0)$ of two strongly Callias-type operators, which are equal outside of a compact set. Even though in our situation the $eta$-invariants of $A_1$ and $A_0$ are not defined, the relative $eta$-invariant behaves as if it were the difference $eta(A_1)-eta(A_0)$. We also define the spectral flow of a family of such operators and use it compute the variation of the relative $eta$-invariant.
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