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Some results on curvature and topology of Finsler manifolds

Bing Ye Wu (2013)

Annales Polonici Mathematici

We investigate the curvature and topology of Finsler manifolds, mainly the growth of the fundamental group. By choosing a new counting function for the fundamental group that does not rely on the generators, we are able to discuss the topic in a more general case, namely, we do not demand that the manifold is compact or the fundamental group is finitely generated. Among other things, we prove that the fundamental group of a forward complete and noncompact Finsler n-manifold (M,F) with nonnegative...

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 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...

Currently displaying 61 – 80 of 90