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The Bellaterra connection.

Jaak Peetre (1993)

Publicacions Matemàtiques

A new object is introduced - the "Fischer bundle". It is, formally speaking, an Hermitean bundle of infinite rank over a bounded symmetric domain whose fibers are Hilbert spaces whose elements can be realized as entire analytic functions square integrable with respect to a Gaussian measure ("Fischer spaces"). The definition was inspired by our previous work on the "Fock bundle". An even more general framework is indicated, which allows one to look upon the two concepts from a unified point of view....

The diffeomorphism group of a Lie foliation

Gilbert Hector, Enrique Macías-Virgós, Antonio Sotelo-Armesto (2011)

Annales de l’institut Fourier

We describe explicitly the group of transverse diffeomorphisms of several types of minimal linear foliations on the torus T n , n ≥ 2 . We show in particular that non-quadratic foliations are rigid, in the sense that their only transverse diffeomorphisms are ± Id and translations. The description derives from a general formula valid for the group of transverse diffeomorphisms of any minimal Lie foliation on a compact manifold. Our results generalize those of P. Donato and P. Iglesias for T 2 , P. Iglesias and...

The Frölicher-Nijenhuis bracket on some functional spaces

Ivan Kolář, Marco Modungo (1998)

Annales Polonici Mathematici

Two fiber bundles E₁ and E₂ over the same base space M yield the fibered set ℱ(E₁,E₂) → M, whose fibers are defined as C ∞ ( E ₁ ₓ , E ₂ ₓ ) , for each x ∈ M. This fibered set can be regarded as a smooth space in the sense of Frölicher and we construct its tangent prolongation. Then we extend the Frölicher-Nijenhuis bracket to projectable tangent valued forms on ℱ(E₁,E₂). These forms turn out to be a kind of differential operators. In particular, we consider a general connection on ℱ(E₁,E₂) and study the associated...

The garden of quantum spheres

Ludwik Dąbrowski (2003)

Banach Center Publications

A list of known quantum spheres of dimension one, two and three is presented.

The geometry of Kato Grassmannians

Bogdan Bojarski, Giorgi Khimshiashvili (2005)

Open Mathematics

We discuss Fredholm pairs of subspaces and associated Grassmannians in a Hilbert space. Relations between several existing definitions of Fredholm pairs are established as well as some basic geometric properties of the Kato Grassmannian. It is also shown that the so-called restricted Grassmannian can be endowed with a natural Fredholm structure making it into a Fredholm Hilbert manifold.

The Lie group of real analytic diffeomorphisms is not real analytic

Rafael Dahmen, Alexander Schmeding (2015)

Studia Mathematica

We construct an infinite-dimensional real analytic manifold structure on the space of real analytic mappings from a compact manifold to a locally convex manifold. Here a map is defined to be real analytic if it extends to a holomorphic map on some neighbourhood of the complexification of its domain. As is well known, the construction turns the group of real analytic diffeomorphisms into a smooth locally convex Lie group. We prove that this group is regular in the sense of Milnor. ...

The Morse-Sard-Brown Theorem for Functionals on Bounded Fréchet-Finsler Manifolds

Kaveh Eftekharinasab (2015)

Communications in Mathematics

In this paper we study Lipschitz-Fredholm vector fields on bounded Fréchet-Finsler manifolds. In this context we generalize the Morse-Sard-Brown theorem, asserting that if M is a connected smooth bounded Fréchet-Finsler manifold endowed with a connection 𝒦 and if ξ is a smooth Lipschitz-Fredholm vector field on M with respect to 𝒦 which satisfies condition (WCV), then, for any smooth functional l on M which is associated to ξ , the set of the critical values of l is of first category in ℝ . Therefore,...

The rectifiable distance in the unitary Fredholm group

Esteban Andruchow, Gabriel Larotonda (2010)

Studia Mathematica

Let U c ( ) = u: u unitary and u-1 compact stand for the unitary Fredholm group. We prove the following convexity result. Denote by d ∞ the rectifiable distance induced by the Finsler metric given by the operator norm in U c ( ) . If u ₀ , u ₁ , u ∈ U c ( ) and the geodesic β joining u₀ and u₁ in U c ( ) satisfy d ∞ ( u , β ) < π / 2 , then the map f ( s ) = d ∞ ( u , β ( s ) ) is convex for s ∈ [0,1]. In particular, the convexity radius of the geodesic balls in U c ( ) is π/4. The same convexity property holds in the p-Schatten unitary groups U p ( ) = u: u unitary and u-1 in the p-Schatten class...

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