arXiv research

Introduces infinite-dimensional differential geometry using Bastiani calculus.

This study describes the Fisher-Rao metric on Gaussian measures in infinite-dimensional spaces.

Shape analysis and compuational anatomy both make use of sophisticated tools from infinite-dimensional differential manifolds and Riemannian geometry on spaces of functions. While comprehensive references for the mathematical foundations exist, it is sometimes difficult to gain an overview how differential geometry and…

We describe the exponential map from an infinite-dimensional Lie algebra to an infinite-dimensional group of operators on a Hilbert space. Notions of differential geometry are introduced for these groups. In particular, the Ricci curvature, which is understood as the limit of the Ricci curvature of finite-dimensional g…

Study on the geometry of spacelike hypersurfaces in spacetime.

Workshop on shape analysis discusses new research directions.

We lay foundations of the subject in the title, on which we build in another paper devoted to isometries in spaces of Kähler metrics.

In this paper we will investigate the global properties of complete Hilbert manifolds with upper and lower bounded sectional curvature. We shall prove the Focal Index Lemma that we will allow us to extend some classical results of finite dimensional Riemannian geometry such as Rauch and Berger Theorems and the Topogono…

In this paper, we define and study sub-Riemannian structures on Banach manifolds. We obtain extensions of the Chow-Rashevski theorem for exact controllability, and give conditions for the existence of a Hamiltonian geodesic flow despite the lack of a Pontryagin Maximum Principle in the infinite dimensional setting.

We present a new construction of tubular neighborhoods in (possibly infinite dimensional) Riemannian manifolds M, which allows us to show that if G is an arbitrary group acting isometrically on M, then every G-invariant submanifold with locally trivial normal bundle has a G-invariant total tubular neighborhood. We appl…

Let $M$, $N$ be finite-dimensional manifolds with $M$ compact. This paper looks at the Riemnannian geometry on the space $C^\infty(M,N)$ of smooth maps equipped with the $L^2$-Riemannian metric. This metric was used by Ebin and Marsden in the proof of the well-posedness of the incompressible Euler equation and is relat…

We generalize the concept of sub-Riemannian geometry to infinite-dimensional manifolds modeled on convenient vector spaces. On a sub-Riemannian manifold $M$, the metric is defined only on a sub-bundle $\calH$ of the tangent bundle $TM$, called the horizontal distribution. Similarly to the finite-dimensional case, we ar…

Riemannian stochastic gradient descent approximates a diffusion process called Riemannian stochastic modified flow.

Improved semialgebraic choices with linear complexity.

The paper proves Sard's theorem for polynomial maps in infinite dimensions.

Infinite-dimensional geometry: completeness and geodesics in Hilbert manifolds.

Geometric framework for Milnor classifying spaces in diffeological spaces.

Consider the sum of the first $N$ eigenspaces for the Laplacian on a Riemannian manifold. A basis for this space determines a map to Euclidean space and for $N$ sufficiently large the map is an embedding. In analogy with a fruitful idea of Kähler geometry, we define (Riemannian) Bergman metrics of degree $N$ to be thos…

Extends metrics for SPD matrices to infinite dimensions.

Introduces new geometric framework for probability densities on manifolds.

Research covers geometry, analysis, and integration on infinite-dimensional spaces.

The space of probability densities is an infinite-dimensional Riemannian manifold, with Riemannian metrics in two flavors: Wasserstein and Fisher--Rao. The former is pivotal in optimal mass transport (OMT), whereas the latter occurs in information geometry---the differential geometric approach to statistics. The Rieman…

Infinite-dimensional contact geometry explored.

Diffeologies unify infinite-dimensional geometry and PDEs, enhancing classical function spaces.

Using symplectic techniques and spectral analysis of smooth paths of self-adjoint operators, we characterize the set of conjugate instants along a geodesic in an infinite dimensional Riemannian Hilbert manifold.

Hodge theory is a beautiful synthesis of geometry, topology, and analysis, which has been developed in the setting of Riemannian manifolds. On the other hand, spaces of images, which are important in the mathematical foundations of vision and pattern recognition, do not fit this framework. This motivates us to develop …

We study the geometry of the space of densities $\VolM$, which is the quotient space $\Diff(M)/\Diff_μ(M)$ of the diffeomorphism group of a compact manifold $M$ by the subgroup of volume-preserving diffemorphisms, endowed with a right-invariant homogeneous Sobolev $\dot{H}^1$-metric. We construct an explicit isometry f…

Develops Hilbert geometries and characterizes their isometries.

In infinite dimensional Heisenberg group, degenerate distances linked to unbounded curvature.

Develops a new algebraic framework for differential geometry of infinite dimensional spaces.

A classical result in Riemannian geometry states that the absolutely continuous curves into a (finite-dimensional) Riemannian manifold form an infinite-dimensional manifold. In the present paper this construction and related results are generalised to absolutely continuous curves with values in a strong Riemannian mani…

Optimizes shapes on non-standard manifolds.

Study validates Michor-Mumford conjecture in infinite dimensional Hilbertian H-type groups.

Geodesics become an essential element of the geometry of a semi-Riemannian manifold. In fact, their differences and similarities with the (positive definite) Riemannian case, constitute the first step to understand semi-Riemannian Geometry. The progress in the last two decades has become impressive, being especially re…

Let (M,g) be a compact, connected and oriented Riemannian manifold. We denote D the space of smooth probability density functions on M. In this paper, we show that the Frechet manifold D is equipped with a Riemannian metric g^{D} and an affine connection \nabla^{D} which are infinite dimensional analogues of the Fisher…

This paper opens the series of articles supplemental to the series (hep-th/9405050,q-alg/9610026,q-alg/9611003,q-alg/9611019,funct-an/9611003), which also lies in lines of general ideology exposed in the review (mp_arc/96-477). The main purpose of the activity, which has its origin and motivation presumably in the auth…

Riemannian symmetric spaces are fundamental objects in finite dimensional differential geometry. An important problem is the construction of symmetric spaces for generalizations of simple Lie groups, especially their closest infinite dimensional analogues known as Kac-Moody groups. We solve this problem and construct a…

In this paper we show that if one writes down the structure equations for the evolution of a curve embedded in an (n)-dimensional Riemannian manifold with constant curvature this leads to a symplectic, a Hamiltonian and an hereditary operator. This gives us a natural connection between finite dimensional geometry, infi…

We prove a semi-Riemannian version of the celebrated Morse Index Theorem for geodesics in semi-Riemannian manifolds; we consider the general case of both endpoints variable on two submanifolds. The key role of the theory is played by the notion of the {\em Maslov index} of a semi-Riemannian geodesic, which is a homolog…

We consider a natural Riemannian metric on the infinite dimensional manifold of all embeddings from a manifold into a Riemannian manifold, and derive its geodesic equation in the case $\Emb(\Bbb R,\Bbb R)$ which turns out to be Burgers' equation. Then we derive the geodesic equation, the curvature, and the Jacobi equat…

Generalized Stacey-Roberts lemma for Banach manifolds.

Periodic geodesics on Hilbert half-Lie groups exist whenever the fundamental group is nontrivial.

We show that the standard picture regarding the notion of stability of constant scalar curvature metrics in Kähler geometry described by S.K. Donaldson, which involves the geometry of infinite-dimensional groups and spaces, can be applied to the constant scalar curvature metrics in Sasaki geometry with only few modific…

The study defines divergence for multivector fields on infinite-dimensional manifolds.

The paper extends Cartan development to infinite dimensional Lie groups.

We study heat kernel measures on sub-Riemannian infinite-dimensional Heisenberg-like Lie groups. In particular, we show that Cameron-Martin type quasi-invariance results hold in this subelliptic setting and give $L^p$-estimates for the Radon-Nikodym derivatives. The main ingredient in our proof is a generalized curvatu…

Introduces a new geometric framework for probability distributions.

This work presents a parametrized family of divergences, namely Alpha-Beta Log- Determinant (Log-Det) divergences, between positive definite unitized trace class operators on a Hilbert space. This is a generalization of the Alpha-Beta Log-Determinant divergences between symmetric, positive definite matrices to the infi…