Infinite dim. Riemannian geometry: theory and computations

A Riemannian Manifold is a space where it is possible to measure lengths and angles: the distance between two points is then defined as the length of the shortest path connecting these points. These length minimizing paths are also called geodesics and are described by the geodesic equation. The simplest example of a Riemannian manifold is the Euclidean space, where geodesics are exactly straight lines. Consequently this is a flat space. There are, however, also curved spaces such as the sphere or the torus. The classic theory of Riemannian geometry considers situations where the underlying manifold is a finite dimensional space. Nevertheless Bernhard Riemann mentioned the possibility of infinite dimensional Riemannian manifolds already in his Habilitationsschrift, which is regarded as the birth place of Riemannian geometry: There are manifoldnesses in which the determination of position requires not a finite number, but either an endless series or a continuous manifoldness of determinations of quantity. Such manifoldnesses are, for example, the possible determinations of a function for a given region, the possible shapes of a solid figure, etc. (Quote taken from the translation of Riemann`s Habilitationsschrift) The last example which Riemann mentions is the central motivation of the present research project: using Riemannian geometry to study similarities between geometric shapes. This area, which is also called shape analysis, has a multitude of different applications in areas such as medical informatics, computer graphics and data science. The biggest challenge for the application of Riemannian methods in shape analysis is the infinite dimensionality of the space of all geometric shapes: many classical results of finite dimensional Riemannian geometry might fail in this infinite dimensional setting and several inconvenient pathologies can arise. This observation is the starting point of the first part of this research project, where we aim to prove the theorem of Hopf-Rinow, a classical result in finite dimensional Riemannian Geometry, for a particular class of infinite dimensional manifolds; it is well known that this result fails for general infinite dimensional manifolds In the second part of the research project we consider a particular shape space: the space of all surfaces. This space is particular important in applications such as in the analysis of (human) organs in medical informatics. Statistics on the infinite dimensional space of surfaces is based on a notion of distance and angles in the Riemannian sense. For the practical application we need to implement these methods on a computer. This challenge forms the second part of my project.

A Riemannian Manifold is a space where it is possible to measure lengths and angles: the distance between two points is then defined as the length of the shortest path connecting these points. These length minimizing paths are also called geodesics and are described by the geodesic equation. The simplest example of a Riemannian manifold is the Euclidean space, where geodesics are exactly straight lines. Consequently this is a flat space. The classic theory of Riemannian geometry considers situations where the underlying manifold is a finite dimensional space. Nevertheless Bernhard Riemann mentioned the possibility of infinite dimensional Riemannian manifolds already in his Habilitationsschrift, which is regarded as the birth place of Riemannian geometry: "There are manifoldnesses in which the determination of position requires not a finite number, but either an endless series or a continuous manifoldness of determinations of quantity. Such manifoldnesses are, for example, the possible determinations of a function for a given region, the possible shapes of a solid figure, etc." (Quote taken from the translation of Riemann's Habilitationsschrift) The last example which Riemann mentions is the central motivation of the present research project: using Riemannian geometry to study similarities between geometric shapes. This area, which is also called shape analysis, has a multitude of different applications in areas such as medical informatics, computer graphics and data science. The biggest challenge for the application of Riemannian methods in shape analysis is the infinite dimensionality of the space of all geometric shapes: many classical results of finite dimensional Riemannian geometry might fail in this infinite dimensional setting and several inconvenient pathologies can arise. This observation was the starting point of the first part of this research project, where we proved the theorem of Hopf-Rinow, a classical result in finite dimensional Riemannian Geometry, for a particular class of infinite dimensional manifolds; it is well known that this result fails for general infinite dimensional manifolds. This research led to a comprehensive manuscript, which has been recently accepted for publications in the prestigious Journal of the European Mathematical Society. In the second part of the research project we considered a particular shape space: the space of all surfaces. This space is particular important in applications such as in the analysis of (human) organs in medical informatics. Statistics on the infinite dimensional space of surfaces is based on a notion of distance and angles in the Riemannian sense. In the second part of this research project we developed a comprehensive numerical framework for the statistical shape analysis of surfaces with specific applications to the space of human body motions. This part of the research has led to several publications in leading outlets, including a publication in the International Journal of Computer Vision.