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On convergence of intrinsic volumes of Riemannian manifolds

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Let \(\pi :M\rightarrow B\) be a Riemannian submersion of two compact smooth Riemannian manifolds, B is connected. Let \(M(\varepsilon )\) denote the manifold M equipped with the new Riemannian metric which is obtained from the original one by multiplying by \(\varepsilon \) along the vertical subspaces (i.e. along the fibers) and keeping unchanged along the (orthogonal to them) horizontal subspaces. Let \(V_i(M(\varepsilon ))\) denote the ith intrinsic volume. The main result of this note says that \(\lim _{\varepsilon \rightarrow +0}V_i(M(\varepsilon ))=\chi (Z) V_i(B)\) where \(\chi (Z)\) denotes the Euler characteristic of a fiber of \(\pi \).

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Acknowledgements

Part of this work was done during my sabbatical stay at the Kent State University in the academic year 2018/19. I am grateful to this institution for hospitality.

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  1. Department of Mathematics, Tel Aviv University, 69978, Ramat Aviv, Tel Aviv, Israel

    Semyon Alesker

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  1. Semyon Alesker
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Correspondence to Semyon Alesker.

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Partially supported by ISF Grant 865/16 and the US - Israel BSF Grant 2018115.

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Alesker, S. On convergence of intrinsic volumes of Riemannian manifolds. J. Geom. 113, 23 (2022). https://doi.org/10.1007/s00022-022-00634-6

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  • DOI: https://doi.org/10.1007/s00022-022-00634-6

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