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Peter W. Michor

Publications and source records attributed to Peter W. Michor.

At least 19 recordsLinked to original sources
arXiv2026

Landmark shape spaces with induced metrics

We present a unification of Kendall's landmark shape spaces, where rigid motions are factored out and scale fixed on landmark configurations equipped with Euclidean geometry, with landmark configuration spaces carrying Riemannian metrics descending from right-invariant Sobolev metrics on the diffeomorphism group. The resulting new landmark shape spaces achieve the defining properties of both approaches: The regularity of the descending metric prevents landmarks from colliding, the metric is defined in the ambient space independent of the number of landmarks, local rigid transformations are preserved, global rigid motions are removed, and scale fixed. To achieve this, we define a particular Sobolev-type operator, the screened elasticity operator, whose null-space consists exactly of the rigid motions, we show how this operator descends to achieve the desired geometry, and we present approaches to solving matching problems and computing geodesics numerically. The resulting construction allows the use of landmark configuration spaces with sufficiently regular metrics in applications while retaining the shape invariances that are a hallmark of Kendall's shape spaces.

cs.CV↗
arXiv2026

The Wasserstein--Ebin Metric: A Geometric Lift of Unbalanced Optimal Transport to the space of Riemannian metrics

We introduce dynamic and static formulations that formally extend unbalanced optimal transport from the space of positive densities to the space of Riemannian metrics. The first construction is based on a dynamic variational formulation in which the evolution of a Riemannian metric is driven by transport together with a source term. Choosing the $L^2$-metric to penalize the transport vector field and the Ebin metric to penalize the source component yields a new Riemannian metric on the manifold of Riemannian metrics, which we call the Wasserstein--Ebin metric. Our main result shows that the volume map defines a Riemannian submersion from the Wasserstein--Ebin metric to the Wasserstein--Fisher--Rao metric on the space of smooth densities. In addition, we construct a Riemannian submersion from the automorphism group of the tangent bundle onto the space of Riemannian metrics, providing a generalization of Otto's geometric description for the Wasserstein metric to the setting of the Wasserstein--Ebin metric. To propose a static formulation of unbalanced optimal Riemannian metric transport, we introduce two Kullback--Leibler-type divergences on the space of Riemannian metrics: one inspired by matrix information geometry, and another related, through the volume map, to the classical Kullback--Leibler divergence on densities. Establishing a link between the static and dynamic formulations remains an open direction for future work.

math.DG↗
arXiv2026

Symplectic structures on the space of space curves

We present symplectic structures on the shape space of unparameterized space curves that generalize the classical Marsden-Weinstein structure. Our method integrates the Liouville 1-form of the Marsden-Weinstein structure with Riemannian structures that have been introduced in mathematical shape analysis. We also derive Hamiltonian vector fields for several classical Hamiltonian functions with respect to these new symplectic structures.

math.SG↗
arXiv2025

Poisson bivectors on infinite dimensional manifolds

We show that, on a smoothly paracompact convenient manifold $M$ modeled on a convenient space with the bornological approximation property, the dual map of a Poisson bracket factors as a smooth section of the vector bundle $L_{skew}^2(T^*M,\mathbb R)$.

math.DG↗
arXiv2025

Lie derivatives of sections of natural vector bundles

Time derivatives of pullbacks and push forwards along smooth curves of diffeomorphism of sections of natural vector bundles are computed in terms of Lie derivatives along adapted non-autonomous vector fields by extending a key lemma in [Markus Mauhart, Peter W. Michor: Commutators of flows and fields. Arch. Math. (Brno) 28 (1992), 228-236. arXiv:math/9204221]. There is also the analogous result about the first non-vanishing derivative of higher order.

math.DG↗
arXiv2025

The Schouten-Nijenhuis bracket in infinite dimensions

The Schouten-Nijenhuis bracket on smooth infinite-dimensional manifolds $M$ is developed in two steps: For summable multivector fields whose pointwise dual are all differential form, and in an extended form for multivector fields which are sections of $L^{\bullet}_{\text{skew}}(T^*M,\mathbb R)$. We need to either assume that $C^{\infty}(M)$ separates points on $TM$, or consider sheaves of local sections.

math.DG↗
arXiv2025

Regularity and completeness of half-Lie groups

Half Lie groups exist only in infinite dimensions: They are smooth manifolds and topological groups such that right translations are smooth, but left translations are merely required to be continuous. The main examples are groups of $H^s$ or $C^k$ diffeomorphisms and semidirect products of a Lie group with kernel an infinite dimensional representation space. Here, we investigate mainly Banach half-Lie groups, the groups of their $C^k$-elements, extensions, and right invariant strong Riemannian metrics on them: surprisingly the full Hopf--Rinow theorem holds, which is not the case in general even for Hilbert manifolds.

math.DG↗
arXiv2024

Geometry of infinite dimensional Cartan Developments

The Cartan development takes a Lie algebra valued 1-form satisfying the Maurer-Cartan equation on a simply connected manifold $M$ to a smooth mapping from $M$ into the Lie group. In this paper this is generalized to infinite dimensional $M$ for infinite dimensional regular Lie groups. The Cartan development is viewed as a generalization of the evolution map of a regular Lie group. The tangent mapping of a Cartan development is identified as another Cartan development.

math.DG↗
arXiv2024

Sobolev metrics on spaces of manifold valued curves

We study completeness properties of reparametrization invariant Sobolev metrics of order $n\ge 2$ on the space of manifold valued open and closed immersed curves. In particular, for several important cases of metrics, we show that Sobolev immersions are metrically and geodesically complete (thus the geodesic equation is globally well-posed). These results were previously known only for closed curves with values in Euclidean space. For the class of constant coefficient Sobolev metrics on open curves, we show that they are metrically incomplete, and that this incompleteness only arises from curves that vanish completely (unlike "local" failures that occur in lower order metrics).

math.DG↗
arXiv2023

Geometry of Sample Spaces

In statistics, independent, identically distributed random samples do not carry a natural ordering, and their statistics are typically invariant with respect to permutations of their order. Thus, an $n$-sample in a space $M$ can be considered as an element of the quotient space of $M^n$ modulo the permutation group. The present paper takes this definition of sample space and the related concept of orbit types as a starting point for developing a geometric perspective on statistics. We aim at deriving a general mathematical setting for studying the behavior of empirical and population means in spaces ranging from smooth Riemannian manifolds to general stratified spaces. We fully describe the orbifold and path-metric structure of the sample space when $M$ is a manifold or path-metric space, respectively. These results are non-trivial even when $M$ is Euclidean. We show that the infinite sample space exists in a Gromov-Hausdorff type sense and coincides with the Wasserstein space of probability distributions on $M$. We exhibit Fréchet means and $k$-means as metric projections onto 1-skeleta or $k$-skeleta in Wasserstein space, and we define a new and more general notion of polymeans. This geometric characterization via metric projections applies equally to sample and population means, and we use it to establish asymptotic properties of polymeans such as consistency and asymptotic normality.

math.ST↗
arXiv2022

Orbifold-like and proper $\mathfrak g$-manifolds

In [4] and [5], we generalized the concept of completion of an infinitesimal group action $ζ: {\mathfrak g} \to \mathfrak X (M)$ to an actual group action on a (non-compact) manifold $M$, originally introduced by R. Palais [9], and showed by examples that this completion may have quite pathological properties (much like the leaf space of a foliation). In the present paper, we introduce and investigate a tamer class of $\mathfrak g$-manifolds, called orbifold--like, for which the completion has an orbifold structure. This class of $\mathfrak g$-manifolds is reasonably well-behaved with respect to its local topological and smooth structure to allow for many geometric constructions to make sense. In particular, we investigate proper $\mathfrak g$-actions and generalize many of the usual properties of proper group actions to this more general setting.

math.DG↗
arXiv2021

Smooth perturbations of the functional calculus and applications to Riemannian geometry on spaces of metrics

We show for a certain class of operators $A$ and holomorphic functions $f$ that the functional calculus $A\mapsto f(A)$ is holomorphic. Using this result we are able to prove that fractional Laplacians $(1+Δ^g)^p$ depend real analytically on the metric $g$ in suitable Sobolev topologies. As an application we obtain local well-posedness of the geodesic equation for fractional Sobolev metrics on the space of all Riemannian metrics.

math.DG↗
arXiv2020

Manifolds of mappings for continuum mechanics

This is an overview article. After an introduction to convenient calculus in infinite dimensions, the foundational material for manifolds of mappings is presented. The central character is the smooth convenient manifold $C^{\infty}(M,N)$ of all smooth mappings from a finite dimensional Whitney manifold germ $M$ into a smooth manifold $N$. A Whitney manifold germ is a smooth (in the interior) manifold with a very general boundary, but still admitting a continuous Whitney extension operator. This notion is developed here for the needs of geometric continuum mechanics.

math.DG↗
arXiv2020

Fractional Sobolev metrics on spaces of immersions

We prove that the geodesic equations of all Sobolev metrics of fractional order one and higher on spaces of diffeomorphisms and, more generally, immersions are locally well posed. This result builds on the recently established real analytic dependence of fractional Laplacians on the underlying Riemannian metric. It extends several previous results and applies to a wide range of variational partial differential equations, including the well-known Euler-Arnold equations on diffeomorphism groups as well as the geodesic equations on spaces of manifold-valued curves and surfaces.

math.DG↗
arXiv2019

Closed surfaces with different shapes that are indistinguishable by the SRNF

The Square Root Normal Field (SRNF), introduced by Jermyn et al. in [3], provides a way of representing immersed surfaces in $\mathbb R^3$, and equipping the set of these immersions with a "distance function" (to be precise, a pseudometric) that is easy to compute. Importantly, this distance function is invariant under reparametrizations (i.e., under self-diffeomorphisms of the domain surface) and under rigid motions of $\mathbb R^3$. Thus, it induces a distance function on the shape space of immersions, i.e., the space of immersions modulo reparametrizations and rigid motions of $\mathbb R^3$. In this paper, we give examples of the degeneracy of this distance function, i.e., examples of immersed surfaces (some closed and some open) that have the same SRNF, but are not the same up to reparametrization and rigid motions. We also prove that the SRNF does distinguish the shape of a standard sphere from the shape of any other immersed surface, and does distinguish between the shapes of any two embedded strictly convex surfaces.

math.DG↗
arXiv2018

Geometry of the Fisher-Rao metric on the space of smooth densities on a compact manifold

It is known that on a closed manifold of dimension greater than one, every smooth weak Riemannian metric on the space of smooth positive densities that is invariant under the action of the diffeomorphism group, is of the form $$ G_μ(α,β)=C_1(μ(M)) \int_M \fracαμ\fracβμ\,μ+ C_2(μ(M)) \int_Mα\cdot \int_Mβ$$ for some smooth functions $C_1,C_2$ of the total volume $μ(M)$. Here we determine the geodesics and the curvature of this metric and study geodesic and metric completeness.

math.DG↗
arXiv2018

Moser's theorem on manifolds with corners

Moser's theorem (1965) states that the diffeomorphism group of a compact manifold acts transitively on the space of all smooth positive densities with fixed volume. Here we describe the extension of this result to manifolds with corners. In particular we obtain Moser's theorem on simplices. The proof is based on Banyaga's paper (1974), where Moser's theorem is proven for manifolds with boundary. A cohomological interpretation of Banyaga's operator is given, which allows a proof of Lefschetz duality using differential forms.

math.DG↗