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1. Integrated Construction of Multimodal Atlases with Structural Connectomes in the Space of Riemannian Metrics.

   Campbell, Kristen M.; Dai, Haocheng; Su, Zhe; Bauer, Martin; Fletcher, Thomas P.; Joshi, Sarang C. (July 2022, The journal of machine learning for biomedical imaging)

   The structural network of the brain, or structural connectome, can be represented by fiber bundles generated by a variety of tractography methods. While such methods give qualitative insights into brain structure, there is controversy over whether they can provide quantitative information, especially at the population level. In order to enable population-level statistical analysis of the structural connectome, we propose representing a connectome as a Riemannian metric, which is a point on an infinite-dimensional manifold. We equip this manifold with the Ebin metric, a natural metric structure for this space, to get a Riemannian manifold along with its associated geometric properties. We then use this Riemannian framework to apply object-oriented statistical analysis to define an atlas as the Fréchet mean of a population of Riemannian metrics. This formulation ties into the existing framework for diffeomorphic construction of image atlases, allowing us to construct a multimodal atlas by simultaneously integrating complementary white matter structure details from DWMRI and cortical details from T1-weighted MRI. We illustrate our framework with 2D data examples of connectome registration and atlas formation. Finally, we build an example 3D multimodal atlas using T1 images and connectomes derived from diffusion tensors estimated from a subset of subjects from the Human Connectome Project. 

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2. A relaxed approach for curve matching with elastic metrics

   https://doi.org/10.1051/cocv/2018053  

   Bauer, Martin; Bruveris, Martins; Charon, Nicolas; Møller-Andersen, Jakob (January 2019, ESAIM: Control, Optimisation and Calculus of Variations)

   In this paper, we study a class of Riemannian metrics on the space of unparametrized curves and develop a method to compute geodesics with given boundary conditions. It extends previous works on this topic in several important ways. The model and resulting matching algorithm integrate within one common setting both the family of H 2 -metrics with constant coefficients and scale-invariant H 2 -metrics on both open and closed immersed curves. These families include as particular cases the class of first-order elastic metrics. An essential difference with prior approaches is the way that boundary constraints are dealt with. By leveraging varifold-based similarity metrics we propose a relaxed variational formulation for the matching problem that avoids the necessity of optimizing over the reparametrization group. Furthermore, we show that we can also quotient out finite-dimensional similarity groups such as translation, rotation and scaling groups. The different properties and advantages are illustrated through numerical examples in which we also provide a comparison with related diffeomorphic methods used in shape registration. 

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3. Barcode entropy of geodesic flows

   https://doi.org/10.4171/JEMS/1572  

   Ginzburg, Viktor L; Gürel, Başak Z; Mazzucchelli, Marco (December 2024, Journal of the European Mathematical Society)

   We introduce and study the barcode entropy for geodesic flows of closed Riemannian manifolds, which measures the exponential growth rate of the number of not-too-short bars in the Morse-theoretic barcode of the energy functional. We prove that the barcode entropy bounds from below the topological entropy of the geodesic flow and, conversely, bounds from above the topological entropy of any hyperbolic compact invariant set. As a consequence, for Riemannian metrics on surfaces, the barcode entropy is equal to the topological entropy. A key to the proofs and of independent interest is a crossing energy theorem for gradient flow lines of the energy functional. 

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4. Almost isotropy-maximal manifolds of non-negative curvature

   https://doi.org/10.1090/9100  

   Dong, Zheting; Escher, Christine; Searle, Catherine (May 2024, Transactions of the American Mathematical Society)

   We extend the equivariant classification results of Escher and Searle for closed, simply connected, Riemannian $n$-manifolds with non-negative sectional curvature admitting isometric isotropy-maximal torus actions to the class of such manifolds admitting isometric strictly almost isotropy-maximal torus actions. In particular, we prove that any such manifold is equivariantly diffeomorphic to the free, linear quotient by a torus of a product of spheres of dimensions greater than or equal to three. 

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5. On Eisenhart’s Type Theorem for Sub-Riemannian Metrics on Step $$2$$ Distributions with $$\mathrm{ad}$$-Surjective Tanaka Symbols

   https://doi.org/10.1134/S1560354724020023  

   Lin, Zaifeng; Zelenko, Igor (February 2024, Regular and Chaotic Dynamics)

   The classical result of Eisenhart states that, if a Riemannian metric admits a Riemannian metric that is not constantly proportional to and has the same (parameterized) geodesics as in a neighborhood of a given point, then is a direct product of two Riemannian metrics in this neighborhood. We introduce a new generic class of step graded nilpotent Lie algebras, called -surjective, and extend the Eisenhart theorem to sub-Riemannian metrics on step distributions with -surjective Tanaka symbols. The class of ad-surjective step nilpotent Lie algebras contains a well-known class of algebras of H-type as a very particular case. 

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