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On the space of riemannian metrics satisfying surgery stable curvature conditions

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Abstract

We utilize a condition for algebraic curvature operators called surgery stability as suggested by the work of Hoelzel to investigate the space of riemannian metrics over closed manifolds satisfying these conditions. Our main result is a parametrized Gromov–Lawson construction with not necessarily trivial normal bundles and shows that the homotopy type of this space of metrics is invariant under surgeries of a suitable codimension. This is a generalization of a well-known theorem by Chernysh and Walsh for metrics of positive scalar curvature. As an application of our method, we show that the space of metrics of positive scalar curvature on quaternionic projective spaces are homotopy equivalent to that on spheres.

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Notes

  1. Precisely as in the scalar curvature case, note that \({\mathcal {R}}^{{{\,\textrm{torp}\,}}}_C(M_0)\) and \({\mathcal {R}}^{{{\,\textrm{torp}\,}}}_C(M_1)\) are homeomorphic and thus Theorem A yields a chain of weak homotopy equivalences. It can be strengthened to a homotopy equivalence using Whitehead’s theorem because the space of metrics satisfying C (an open condition) is an open subset of the space of metrics, which was shown by Palais [25, Theorem 14] to be dominated by CW-complexes.

  2. e.g. [4, Proposition 3.4].

  3. This condition is sometimes also called positive Einstein or the metric is said to have positive Einstein tensor. We will refrain from these terms to avoid confusion with Einstein metrics with positive Einstein constant.

  4. As common notation suggests, we will write relations such as \(\sec (R) < \alpha \) to mean “\(\sec (R, E) < \alpha \) for every plane \(E < {\mathbb {E}}^n\)”.

  5. Smoothing here means to convolute with a Dirac delta centered at \(\frac{\delta \pi }{2}\) to make sure the derivatives of even order in the definition of \(\beta _\delta \) agree on both sides at \(\frac{\delta \pi }{2}\).

  6. While there have been similar ideas prior to the work of Chernysh (e.g. [5, 12]), the formalism as seen in the following is essentially laid out in [6].

  7. Note that in general this cannot be done in a continuous way if \(\nu N\) is not assumed to be trivial.

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Acknowledgements

The author would like to express his gratitude to Prof. W. Tuschmann for his constant support during the preparation of this work, which is based on results of the author’s Ph.D. thesis. Moreover, the author extends thanks to Prof. J. Ebert, Dr. M. Wiemeler and Dr. G. Frenck for helpful discussions on the subject, as well as to the anonymous referee for considerate comments. In particular, I would like to thank Dr. M. Wiemeler for his suggestion to study Chernysh’s construction with non-trivial normal bundles.

Funding

The author was supported by the SNSF (Swiss National Science Foundation) Project 200021E-172469 [as an SNF Postdoc September 2019 – August 2020] and the DFG (Deutsche Forschungsgemeinschaft) Priority programme Geometry at infinity (SPP 2026) [as an associated member February 2019–August 2019]. The author acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 281869850 (RTG 2229) [as an associated member October 2016 – August 2019].

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  1. University of Fribourg, Ch. du musée 23, 1700, Fribourg, Switzerland

    Jan-Bernhard Kordaß

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This work was supported in part by the SNSF-Project 200021E-172469 and the DFG-Priority programme Geometry at infinity (SPP 2026). The author acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 281869850 (RTG 2229).

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Kordaß, JB. On the space of riemannian metrics satisfying surgery stable curvature conditions. Math. Ann. 388, 1841–1878 (2024). https://doi.org/10.1007/s00208-023-02563-4

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