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Frame bundle
About: Frame bundle is a research topic. Over the lifetime, 1600 publications have been published within this topic receiving 23049 citations.
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TL;DR: In this article, the mean topological dimension for random bundle transformations was introduced, and it was shown that continuous bundle random dynamical systems with finite topological entropy or small boundary property have zero topological dimensions.
Abstract: We introduce the mean topological dimension for random bundle transformations, and show that continuous bundle random dynamical systems with finite topological entropy, or the small boundary property have zero mean topological dimensions.
6 citations
01 Jan 2005
TL;DR: Using the notion of Levi form of a smooth distribution, the authors discusses the local and the global problem of existence of one horizontal section of the smooth vector bundle endowed with a horizontal distribution, and proposes a solution to the problem.
Abstract: Using the notion of Levi form of a smooth distribution, we discuss the local and the global problem of existence of one horizontal section of a smooth vector bundle endowed with a horizontal distribution.
6 citations
TL;DR: In this paper, the authors studied global properties of the global (center-)stable manifold of a normally attracting invariant manifold (NAIM) and showed that the global stable foliation has a topological disk bundle structure, and that the dynamics restricted to the stable manifold of compact inflowing NAIM are globally topologically conjugate to the linearized transverse dynamics at the NAIM.
Abstract: We study global properties of the global (center-)stable manifold of a normally attracting invariant manifold (NAIM), the special case of a normally hyperbolic invariant manifold (NHIM) with empty unstable bundle. We show that the global stable foliation of a NAIM has the structure of a topological disk bundle, and that similar statements hold for inflowing NAIMs and for general compact NHIMs. Furthermore, the global stable foliation has a $C^k$ disk bundle structure if the local stable foliation is assumed $C^k$. We then show that the dynamics restricted to the stable manifold of a compact inflowing NAIM are globally topologically conjugate to the linearized transverse dynamics at the NAIM. Moreover, we give conditions ensuring the existence of a global $C^k$ linearizing conjugacy. We illustrate the theory by giving applications to geometric singular perturbation theory in the case of an attracting critical manifold: we show that the domain of the Fenichel Normal Form can be extended to the entire global stable manifold, and under additional nonresonance assumptions we derive a smooth global linear normal form.
6 citations
TL;DR: In this paper, it was shown that every bijective linear isometry between the continuous section spaces of two non-square Banach bundles gives rise to a Banach bundle isomorphism.
Abstract: We show in this paper that every bijective linear isometry between the continuous section spaces of two non-square Banach bundles gives rise to a Banach bundle isomorphism. This is to support our expectation that the geometric structure of the continuous section space of a Banach bundle determines completely its bundle structures. We also describe the structure of an into isometry from a continuous section space into an other. However, we demonstrate by an example that a non-surjective linear isometry can be far away from a subbundle embedding.
6 citations