Topic
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: Two constructions in geometric deep learning are introduced for transporting orientation-dependent convolutional filters over a manifold in a continuous way and thereby defining a convolution operator that naturally incorporates the rotational effect of holonomy.
Abstract: We introduce two constructions in geometric deep learning for 1) transporting orientation-dependent convolutional filters over a manifold in a continuous way and thereby defining a convolution operator that naturally incorporates the rotational effect of holonomy; and 2) allowing efficient evaluation of manifold convolution layers by sampling manifold valued random variables that center around a weighted diffusion mean. Both methods are inspired by stochastics on manifolds and geometric statistics, and provide examples of how stochastic methods -- here horizontal frame bundle flows and non-linear bridge sampling schemes, can be used in geometric deep learning. We outline the theoretical foundation of the two methods, discuss their relation to Euclidean deep networks and existing methodology in geometric deep learning, and establish important properties of the proposed constructions.
4 citations
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TL;DR: In this article, the theory of balance equations of the Continuum Thermodynamics (balance systems) in a geometrical form using Poincare-Cartan formalism of the multisymplectic field theory is presented.
Abstract: In this paper we are presenting the theory of balance equations of the Continuum Thermodynamics (balance systems) in a geometrical form using Poincare-Cartan formalism of the Multisymplectic Field Theory. A constitutive relation $\mathcal{C}$ of a balance system $B_{C}$ is realized as a mapping between a (partial) 1-jet bundle of the configurational bundle $\pi:Y\to X$ and the dual bundle similar to the Legendre mapping of the Lagrangian Field Theory. Invariant (variational) form of the balance system $B_{C}$ is presented in three different forms and the space of admissible variations is defined and studied. Action of automorphisms of the bundle $\pi$ on the constitutive mappings $C$ is studied and it is shown that the symmetry group $Sym(C)$ of the constitutive relation $C$ acts on the space of solutions of the balance system $B_{C}$. Suitable version of Noether Theorem for an action of a symmetry group is presented with the usage of conventional multimomentum mapping. Finally, the geometrical (bundle) picture of the Rational Extended Thermodynamics in terms of Lagrange-Liu fields is developed and the entropy principle is shown to be equivalent to the holonomicy of the current component of the constitutive section.
4 citations
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4 citations
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TL;DR: In this paper, a relation between the Popp measures of sub-Riemannian structures on the total space of a principal bundle and the base manifold is explained. And several concrete cases explicitly.
Abstract: In this note, we explain a relation between the Popp measures of sub-Riemannian structures on the total space of a principal bundle and the base manifold. Then we determine several concrete cases explicitly.
4 citations
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TL;DR: In this paper, the authors construct a lift of an almost complex structure to the cotangent bundle using a connection on the base manifold, which unifies the complete lift defined by Sato and the horizontal lift introduced by Ishihara and Yano.
Abstract: We construct some lift of an almost complex structure to the cotangent bundle, using a connection on the base manifold. This unifies the complete lift defined by Sato and the horizontal lift introduced by Ishihara and Yano. We study some geometric properties of this lift and its compatibility with symplectic forms on the cotangent bundle.
4 citations