Salvato in:
| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2402.02398 |
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Sommario:
- On complete non-compact manifolds with bounded sectional curvature, we consider a class of self-adjoint Dirac-type operators called Dirac-Schrödinger operators. Assuming two Dirac-Schrödinger operators coincide at infinity, by previous work, one can define their relative eta invariant. A typical example of Dirac-Schrödinger operators is the (twisted) spin Dirac operators on spin manifolds which admit a Riemannian metric of uniformly positive scalar curvature. In this case, using the relative eta invariant, we get a geometric formula for the spectral flow on non-compact manifolds, which induces a new proof of Gromov-Lawson's result about compact area enlargeable manifolds in odd dimensions. When two such spin Dirac operators are the boundary restriction of an operator on a manifold with non-compact boundary, under certain conditions, we obtain an index formula involving the relative eta invariant. This generalizes the Atiyah-Patodi-Singer index theorem to non-compact boundary situation. As a result, we can use the relative eta invariant to study the space of uniformly positive scalar curvature metrics on some non-compact connected sums.