arXiv research

The study finds infinitely many different geometries for odd-dimensional manifolds with positive Ricci curvature.

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

Most Finsler metrics have infinite-dimensional holonomy groups.

We construct a infinite-dimensional manifold structure adapted to analytic Lie pseudogroups of infinite type. More precisely, we prove that any isotropy subgroup of an analytic Lie pseudogroup of infinite type is a regular infinite-dimensional Lie group, modelled on a locally convex strict inductive limit of Banach spa…

Proves uniqueness of embedding complex manifold into infinite-dimensional space.

We define submersions f between manifolds M and N modelled on locally convex spaces. If the range N is finite-dimensional or a Banach manifold, then these coincide with the naive notion of a submersion. We study pre-images of submanifolds under submersions and pre-images under mappings whose differentials have dense im…

Study shows how Poisson brackets factor on infinite dimensional manifolds.

Introduces infinite-dimensional differential geometry using Bastiani calculus.

This paper explores topological aspects of index theory for infinite-dimensional manifolds.

The abstract discusses transversality for infinite dimensional manifolds.

Following the approach of Carlet et al.(2011)\cite{CDM}, we construct a class of infinite-dimensional Frobenius manifolds underlying the Toda lattice hierarchy, which are defined on the space of pairs of meromorphic functions with possibly higher-order poles at the origin and at infinity. We also show a connection betw…

The theory of product preserving functors and Weil functors is partly extended to infinite dimensional manifolds, using the theory of $C^\infty$-algebras.

We construct a class of infinite-dimensional Frobenius manifolds on the space of pairs of certain even functions meromorphic inside or outside the unit circle. Via a bi-Hamiltonian recursion relation, the principal hierarchies associated to such Frobenius manifolds are found to be certain extensions of the dispersionle…

Generalized Stacey-Roberts lemma for Banach manifolds.

Existence of balanced embedding proved for complex manifold into infinite-dimensional space.

A manifold's co-invariant cohomology can be infinite-dimensional.

In this paper we propose a new treatment about infinite dimensional manifolds, using the language of category and functor. Our definition of infinite dimensional manifolds is a natural generalization of finite dimensional manifolds in the sense that de Rham cohomology and singular cohomology can be naturally defined an…

Infinite-dimensional contact geometry explored.

Spray-invariant sets maintain geodesics on infinite-dimensional manifolds.

In this article we study properly discontinuous actions on Hilbert manifolds giving new examples of complete Hilbert manifolds with nonnegative, respectively nonpositive, sectional curvature with infinite fundamental group. We also get examples of complete infinite dimensional Kähler manifolds with positive holomorphic…

Book on infinite-dimensional Lie groups, covering basics and various classes.

In this paper we study the homogeneous Kaehler manifolds (h.K.m.) which can be Kaehler immersed into finite or infinite dimensional complex space forms. On one hand we completely classify the h.K.m. which can be Kaehler immersed into a finite or infinite dimensional complex Euclidean or hyperbolic space. Moreover, we e…

Noncompact Ricci-flat solutions have infinite unstable dimensions.

The paper constructs manifolds with infinite holes from a given manifold.

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.

Chern-Weil and Chern-Simons theory extend to certain infinite-rank bundles that appear in mathematical physics. We discuss what is known of the invariant theory of the corresponding infinite-dimensional Lie groups. We use these techniques to detect cohomology classes for spaces of maps between manifolds and for diffeom…

For any integer $n\geq 2$, we construct an infinite family of Stein fillable contact $(4n-1)$-manifolds each of which admits infinitely many pairwise homotopy inequivalent Stein fillings.

Develops Schouten-Nijenhuis bracket on infinite-dimensional manifolds.

The abstract extends deformation and tangent groupoid constructions to infinite-dimensional manifolds.

We describe a relation between the periodic one-dimensional Toda lattice and the quantum cohomology of the periodic flag manifold (an infinite-dimensional Kaehler manifold). This generalizes a result of Givental and Kim relating the open Toda lattice and the quantum cohomology of the finite-dimensional flag manifold. W…

Let $V$ be a separable Hilbert space, possibly infinite dimensional. Let $\St(p,V)$ be the Stiefel manifold of orthonormal frames of $p$ vectors in $V$, and let $\Gr(p,V)$ be the Grassmann manifold of $p$ dimensional subspaces of $V$. We study the distance and the geodesics in these manifolds, by reducing the matter to…

Study finds infinite families of Sasaki-Einstein metrics on spheres.

We introduce a structure of an infinite-dimensional Frobenius manifold on a subspace in the space of pairs of functions analytic inside/outside the unit circle with simple poles at 0/infinity respectively. The dispersionless 2D Toda equations are embedded into a bigger integrable hierarchy associated with this Frobeniu…

The paper constructs infinitely many stably diffeomorphic but non-homotopy equivalent manifolds.

The paper extends Cartan development to infinite dimensional Lie groups.

We give an exhaustive description of all simply connected odd dimensional cohomogeneity one manifolds that can possibly support an invariant metric with positive sectional curvature. Among the known examples of odd dimensional manifolds with positive curvature, apart from spheres, there are two infinite families among …

It is proved that the moduli space of all connected compact orientable embedded minimal affine Lagrangian submanifolds of a complex equiaffine space constitutes an infinite dimensional Frechet manifold (if it is not the empty set). The moduli space of all connected compact orientable metric Lagrangian embedded surfaces…

Universal approximation theorem for differentiable maps on infinite-dimensional manifolds

Study infinite-dimensional Toda manifold at irregular singularity, revealing non-uniqueness of formal solutions.

We present the construction of an infinite dimensional Banach manifold of quantum mechanical states on a Hilbert space H using different types of small perturbations of a given Hamiltonian. We provide the manifold with a flat connection, called the exponential connection, and comment on the possibility of introducing t…

Study fundamental groups of geometric transformation groups using loop spaces.

The paper extends Nambu-Poisson structures to infinite dimensions.

Generalizes neural network approximation to infinite-dimensional manifolds and derivatives.

We prove a Frobenius theorem for Banach distributions on manifolds that are modelled over locally convex spaces. Moreover, we recall how Frobenius theorems can be applied to infinite-dimensional Lie groups and obtain, that given a Lie subalgebra of the Lie algebra of a Lie group that is modelled over a locally convex s…

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

An infinite family of pairwise nonhomeomorphic 13-dimensional positively curved manifolds is constructed

Constructs infinite-dimensional Siegel disc as symplectic and Kaehler quotient.

Develops a mathematical framework for causal fermion systems in infinite dimensions.