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The \(\kappa \)-nullity of Riemannian manifolds and their splitting tensors

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Abstract

We consider Riemannian n-manifolds M with nontrivial \(\kappa \)-nullity “distribution” of the curvature tensor R, namely, the variable rank distribution of tangent subspaces to M where R coincides with the curvature tensor of a space of constant curvature \(\kappa \) (\(\kappa \in \mathbb {R}\)) is nontrivial. We obtain classification theorems under diferent additional assumptions, in terms of low nullity/conullity, controlled scalar curvature or existence of quotients of finite volume. We prove new results, but also revisit previous ones.

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  1. Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão, 1010, São Pulo, SP, 05508-090, Brazil

    Claudio Gorodski & Felippe Guimarães

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  1. Claudio Gorodski
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  2. Felippe Guimarães
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Correspondence to Felippe Guimarães.

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The first named author has been partially supported by the grant number 302882/2017-0 from the National Council for Scientific and Technological Development (CNPq, Brazil) and the project number 2016/23746-6 from the São Paulo Research Foundation (Fapesp, Brazil).

The second named author has been supported by the post-doctoral grant 2019/19494-0 from the São Paulo Research Foundation (Fapesp, Brazil).

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Gorodski, C., Guimarães, F. The \(\kappa \)-nullity of Riemannian manifolds and their splitting tensors. Annali di Matematica 202, 2561–2583 (2023). https://doi.org/10.1007/s10231-023-01330-1

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