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We investigate the geometry of a family of equations in two dimensions which interpolate between the Euler equations of ideal hydrodynamics and the inviscid surface quasi-geostrophic equation. This family can be realised as geodesic equations on groups of diffeomorphisms. We show precisely when the corresponding Riemannian exponential map is non-linear Fredholm of index 0. We further illustrate this by examining the distribution of conjugate points in these settings via a Morse theoretic approach

 

In this article we propose a novel geometric model to study the motion of a physical flag. In our approach, a flag is viewed as an isometric immersion from the square with values in$$\mathbb {R}^3$$$R^3$satisfying certain boundary conditions at the flag pole. Under additional regularity constraints we show that the space of all such flags carries the structure of an infinite dimensional manifold and can be viewed as a submanifold of the space of all immersions. In the second part of the article we equip the space of isometric immersions with its natural kinetic energy and derive the corresponding equations of motion. This approach can be viewed in a spirit similar to Arnold’s geometric picture for the motion of an incompressible fluid.

 

We introduce a family of Finsler metrics, called the$$L^p$$$L^p$-Fisher–Rao metrics$$F_p$$$F_p$, for$$p\in (1,\infty )$$$p\in (1,\infty )$, which generalizes the classical Fisher–Rao metric$$F_2$$$F_2$, both on the space of densities$${\text {Dens}}_+(M)$$${\text{Dens}}_{+}\left(M\right)$and probability densities$${\text {Prob}}(M)$$$\text{Prob}\left(M\right)$. We then study their relations to the Amari–C̆encov$$\alpha $$$\alpha$-connections$$\nabla ^{(\alpha )}$$${\nabla }^{\left(\alpha \right)}$from information geometry: on$${\text {Dens}}_+(M)$$${\text{Dens}}_{+}\left(M\right)$, the geodesic equations of$$F_p$$$F_p$and$$\nabla ^{(\alpha )}$$${\nabla }^{\left(\alpha \right)}$coincide, for$$p = 2/(1-\alpha )$$$p=2/(1-\alpha )$. Both are pullbacks of canonical constructions on$$L^p(M)$$$L^p\left(M\right)$, in which geodesics are simply straight lines. In particular, this gives a new variational interpretation of$$\alpha $$$\alpha$-geodesics as being energy minimizing curves. On$${\text {Prob}}(M)$$$\text{Prob}\left(M\right)$, the$$F_p$$$F_p$and$$\nabla ^{(\alpha )}$$${\nabla }^{\left(\alpha \right)}$geodesics can still be thought as pullbacks of natural operations on the unit sphere in$$L^p(M)$$$L^p\left(M\right)$, but in this case they no longer coincide unless$$p=2$$$p=2$. Using this transformation, we solve the geodesic equation of the$$\alpha $$$\alpha$-connection by showing that the geodesic are pullbacks of projections of straight lines onto the unit sphere, and they always cease to exists after finite time when they leave the positive part of the sphere. This unveils the geometric structure of solutions to the generalized Proudman–Johnson equations, and generalizes them to higher dimensions. In addition, we calculate the associate tensors of$$F_p$$$F_p$, and study their relation to$$\nabla ^{(\alpha )}$$${\nabla }^{\left(\alpha \right)}$.

 

Abstract This paper introduces a set of numerical methods for Riemannian shape analysis of 3D surfaces within the setting of invariant (elastic) second-order Sobolev metrics. More specifically, we address the computation of geodesics and geodesic distances between parametrized or unparametrized immersed surfaces represented as 3D meshes. Building on this, we develop tools for the statistical shape analysis of sets of surfaces, including methods for estimating Karcher means and performing tangent PCA on shape populations, and for computing parallel transport along paths of surfaces. Our proposed approach fundamentally relies on a relaxed variational formulation for the geodesic matching problem via the use of varifold fidelity terms, which enable us to enforce reparametrization independence when computing geodesics between unparametrized surfaces, while also yielding versatile algorithms that allow us to compare surfaces with varying sampling or mesh structures. Importantly, we demonstrate how our relaxed variational framework can be extended to tackle partially observed data. The different benefits of our numerical pipeline are illustrated over various examples, synthetic and real. 

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.