Related papers: Parallel transport on matrix manifolds and Exponen…

The analysis of manifold-valued data requires efficient tools from Riemannian geometry to cope with the computational complexity at stake. This complexity arises from the always-increasing dimension of the data, and the absence of…

If M is a smooth compact connected Riemannian manifold, let P(M) denote the Wasserstein space of probability measures on M. We describe a geometric construction of parallel transport of some tangent cones along geodesics in P(M). We show…

This paper presents a derivation of the parallel transport equation expressed in the Lie algebra of a Lie group endowed with a left-invariant metric.The use of this equation is exemplified on the group of rigid body motions SE(3), using…

We consider the Calder\'on problem for systems with unknown zeroth and first order terms, and improve on previously known results. More precisely, let $(M, g)$ be a compact Riemannian manifold with boundary, let $A$ be a connection matrix…

We show that finite parallel transports of vectors in Riemannian spaces, determined by the multiplication law in the deformed groups of diffeomorphisms, and sequences of infinitesimal parallel transports of vectors along geodesics are…

We study the geodesic orbit property for nilpotent Lie groups $N$ when endowed with a pseudo-Riemannian left-invariant metric. We consider this property with respect to different groups acting by isometries. When $N$ acts on itself by…

Parallel transport is a fundamental tool to perform statistics on Rie-mannian manifolds. Since closed formulae don't exist in general, practitioners often have to resort to numerical schemes. Ladder methods are a popular class of algorithms…

We show that there is an infinite group of special automorphisms of the deformed group of diffeomorphisms, which describes parallel transports in Riemannian spaces of any variable curvature. Generators of translations of such group contain…

Parallel transport is an important step in many discrete algorithms for statistical computing on manifolds. Numerical methods based on Jacobi fields or geodesics parallelograms are currently used in geometric data processing. In this last…

We construct a parallel transport on higher loop spaces of a manifold in term of a higher dimensional generalization of iterated path integrals. Under mild assumptions, we define a de Rham complex on higher loop spaces and we recover a…

We analyze a variational time discretization of geodesic calculus on finite- and certain classes of infinite-dimensional Riemannian manifolds. We investigate the fundamental properties of discrete geodesics, the associated discrete…

In the statistical analysis of shape a goal beyond the analysis of static shapes lies in the quantification of `same' deformation of different shapes. Typically, shape spaces are modelled as Riemannian manifolds on which parallel transport…

We review definitions of generalized parallel transports in terms of Cheeger-Simons differential characters. Integration formulae are given in terms of Deligne-Beilinson cohomology classes. These representations of parallel transport can be…

In this paper we extend recent developments in computational optimal transport to the setting of Riemannian manifolds. In particular, we show how to learn optimal transport maps from samples that relate probability distributions defined on…

We derive three families of orthogonally-equivariant matrix submanifold models for the Grassmann, flag, and Stiefel manifolds respectively. These families are exhaustive -- every orthogonally-equivariant submanifold model of the lowest…

We classify the effective and transitive actions of a Lie group $G$ on an n-dimensional non-degenerate hyperboloid (also called real pseudo-hyperbolic space), under the assumption that $G$ is a closed, connected Lie subgroup of…

The geodesic orbit property is useful and interesting in itself, and it plays a key role in Riemannian geometry. It implies homogeneity and has important classes of Riemannian manifolds as special cases. Those classes include weakly…

Left invariant metrics induced by the p-norms of the trace in the matrix algebra are studied on the general lineal group. By means of the Euler-Lagrange equations, existence and uniqueness of extremal paths for the length functional are…

Kendall shape spaces are a widely used framework for the statistical analysis of shape data arising from many domains, often requiring the parallel transport as a tool to normalise time series data or transport gradient in optimisation…

We consider parallel submanifolds $M$ of a Riemannian symmetric space $N$ and study the question whether $M$ is extrinsically homogeneous in $N$\,, i.e.\ whether there exists a subgroup of the isometry group of $N$ which acts transitively…