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The geometry and structure of compact rank-one ECS manifolds

Terek Couto, Ivo
http://orcid.org/0000-0001-9093-4429
http://rave.ohiolink.edu/etdc/view?acc_num=osu1708352981968645

Abstract Details

2024, Doctor of Philosophy, Ohio State University, Mathematics.
In this thesis, we study essentially conformally symmetric (ECS) manifolds, that is, pseudo-Riemannian manifolds (M,g) with parallel Weyl curvature, which are not locally symmetric or conformally flat. Every ECS manifold carries a distinguished null parallel distribution D (called its Olszak distribution) whose rank, always equal to 1 or 2, is referred to as the rank of (M,g). More precisely, we focus on the rank-one situation: while the local structure of ECS manifolds of either rank was already well-known, little could be said about their global structure. In particular, examples of compact ECS manifolds were only known in dimensions of the form n = 3k + 2, with k ≥ 1, and they are all of rank one, geodesically complete, not locally homogeneous, and diffeomorphic to total spaces of torus bundles over S¹. In Chapter 4, combining analytic and combinatorial methods, we construct compact rank-one ECS manifolds of all dimensions n ≥ 5, and sorted in two distinct classes, called translational and dilational: such dichotomy refers to finiteness or infiniteness of the holonomy group of the natural flat connection induced on D. The translational examples appear in all dimensions n ≥ 5 and are all geodesically complete, without being locally homogeneous. The dilational examples, in turn, could only be produced in odd dimensions n ≥ 5, are all geodesically incomplete, and some of them are locally homogeneous, while others are not. In both cases discussed above, the resulting manifolds were diffeomorphic to total spaces of torus bundles over S¹ . This brings us to Chapter 5, where we show that this topological structure was not accidental: using some ideas from foliation theory, we prove that outside of the locally homogeneous case, and up to passing to a double isometric covering if needed, every compact rank-one ECS manifold must fiber over S¹ in such a way that the leaves of D^⊥ appear as the fibers. Chapter 6 brings the notion of genericity — originally introduced back in Chapter 2 — into play. Roughly speaking, genericity means that the Weyl tensor has a finite centralizer, when acting as a traceless self-adjoint endomorphism of the space V of parallel sections of D^⊥/D is finite; the centralizer being taken relative to O(V). Some of the compact translational examples from Chapter 4 were generic, while others non-generic; the dilational examples were all non-generic. Again, that this was not an accident: we prove that there are no generic dilational compact rank-one ECS manifolds. In particular, as the genericity condition always holds in dimension four, we ultimately conclude that there are no four-dimensional compact rank-one ECS manifolds. The question of existence and classification of compact rank-two ECS manifolds is left for future work.
Andrzej Derdzinski (Advisor)
Jean François Lafont (Committee Member)
Andrey Gogolev (Committee Member)
215 p.
Mathematics
pseudo-Riemannian geometry; parallel Weyl curvature

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Citations

  • Terek Couto, I. (2024). The geometry and structure of compact rank-one ECS manifolds [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1708352981968645

    APA Style (7th edition)

  • Terek Couto, Ivo. The geometry and structure of compact rank-one ECS manifolds. 2024. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1708352981968645.

    MLA Style (8th edition)

  • Terek Couto, Ivo. "The geometry and structure of compact rank-one ECS manifolds." Doctoral dissertation, Ohio State University, 2024. http://rave.ohiolink.edu/etdc/view?acc_num=osu1708352981968645

    Chicago Manual of Style (17th edition)

osu1708352981968645
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