Relative eta invariant and uniformly positive scalar curvature on non-compact manifolds
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910119807156224 |
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| author | Shi, Pengshuai |
| author_facet | Shi, Pengshuai |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_02398 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Relative eta invariant and uniformly positive scalar curvature on non-compact manifolds Shi, Pengshuai Differential Geometry 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. |
| title | Relative eta invariant and uniformly positive scalar curvature on non-compact manifolds |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2402.02398 |