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Some New Approaches to Rigidity Problems in Riemannian Geometry: Lie Groupoids, Poisson Manifolds and Von Neumann Algebras

Hassan Zadeh, Atefeh | 2015

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 47967 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Fanai, Hamid Reza
  7. Abstract:
  8. In this thesis, we study a rigidity problem for a 2-step nilmanifold such as Γ by some information about its geodesic flows, where is a simply connected 2-step nilpotent Lie group with a left invariant metric, and Γ is a discrete cocompact subgroup of . For the solution to this problem, first, we consider an algebraic aspect of it; since isometry groups of simply connected Riemannian manifolds can be characterized in a purely algebraic way, i.e., normalizers. Also, as we will show, proper and smooth actions of Lie groups and closed subgroups of isometries for smooth Riemannian structures can be regarded as the same topic. Then, in a generalized setting, when passing from the case of actions on vector spaces to that of actions on vector bundles, the notations of automorphism and isometry admit straightforward generalizations which however, have rarely been spelled out in the literature. Furthermore, in the expression of vector bundles as the associated fiber bundles to a principal bundle, the notation of groupoids automatically emanates. In this way, not only the rigidity problem will be solved by Lie groupoid notations, but also, the dependence between normalizers to induced automorphisms of them, specially almost inner automorphisms, in Lie groups and Lie algebras will be expressed in terms of centralizers and their cosets. Finally, this process, lead us to another proof of a proposition of Gordon and Mao related to these concepts, too. On the other hand, one noticeable point in the various (but limited) solutions to the rigidity problem is the geodesic flow is significantly stronger than the marked length spectrum on 2-step nilmanifolds. Then, as the second approach, we get a theorem of Gordon, Mao and Schueth about compact 2-step nilmanifolds with symplectically conjugate flows. After analyzing in details of their paper, we could generalize their symplectic rigidity theorem, via symplectic foliations, co-normal submanifolds, momentum maps and Poisson cohomology. Our generalization contains two parts: One part is extension of the symplectic notions to the corresponded Poisson concepts, and the other is 2-step nilmanifolds generalized to manifolds with extensible momentum maps. Finally, for the third approach: first, some analysis propositions applied to rigidity problem was a temptation for us. Then, among functional analysis concepts, we have found that many objects in von Neumann algebras, such as dual pairs, bimodules, tensor products, Morita equivalence and commutants, whose applications iteratively in two before approaches, have direct analogues in the realm of Poisson geometry and groupoids. These links do not seem to exist with ∗-algebras on any types of analytic algebras. In this way, we could recover a Lie group and its action from inside crossed product of a von Neumann algebras. This result to a direct proof of a theorem of O’Neil which was the mainstream of solving our rigidity problem, up to now. Also, we could expose another advanced analytical proof to the proposition of Gordon and Mao which had been proved in the first approach, already. Finally, in the supplement of our progress and in relation to more understanding of geometric and algebraic rule of isometries of a manifold , (), we will show that: The proper action of () on , is specified by giving the orbits of the point of , as the immersed submanifolds. These submanifolds intersect the other submanifolds, i.e, slices at the fixed points of the action, transversally. In this way, by a suitable fibration, can be expressed in terms of the quotients of , () and its slices as the fiber. There is an explicit correspondence relation between their Lie algebras, as well. Lastly, by passing from the special (, ℝ)-equivariant functions, we translate the main tools of the geometry of () to the ones of . These interactions lead us to an exact sequence of vector bundles and Lie algebras sheaves corresponding to a homogeneous bundle, (), which is an example of Atiyah sequence in Lie groupoid theory
  9. Keywords:
  10. Von Neumann Algebra ; Symplectic Manifold ; Lie Groups ; Isometric ; Poisson Geometry

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