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: F-manifolds with a commutative and associative multi-plication on the tangent bundle are called F-Manifolds and are closely related to discrim-inants and Lagrange maps as discussed by the authors.
Abstract: Manifolds with a commutative and associative multi- plication on the tangent bundle are called F-manifolds if a unit field exists and the multiplication satisfies a natural integrability con- dition. They are studied here. They are closely related to discrim- inants and Lagrange maps. Frobenius manifolds are F-manifolds. As an application a conjecture of Dubrovin on Frobenius manifolds and Coxeter groups is proved.
21 citations
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21 citations
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TL;DR: In this article, the authors studied the characteristic exponents of flows in relation with the dynamics of flows on flag bundles and proved a symmetric property of these spectral sets, namely invariance under the Weyl group.
Abstract: This paper studies characteristic exponents of flows in relation with the dynamics of flows on flag bundles. The starting point is a flow on a principal bundle with semi-simple group G. Projection against the Iwasawa decomposition G = KAN defines an additive cocycle over the flow with values in a = logA. Its Lyapunov exponents (limits along trajectories) and Morse exponents (limits along chains) are studied. It is proved a symmetric property of these spectral sets, namely invariance under the Weyl group. It is proved also that these sets are located in certain Weyl chambers, defined from the dynamics on the associated flag bundles. As a special case linear flows on vector bundles are considered.
21 citations
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TL;DR: In this article, the authors discuss the geometric foundation behind the use of stochastic processes in the frame bundle of a smooth manifold to build stochastically models with applications in statistical analysis of non-linear data.
Abstract: We discuss the geometric foundation behind the use of stochastic processes in the frame bundle of a smooth manifold to build stochastic models with applications in statistical analysis of non-linear data. The transition densities for the projection to the manifold of Brownian motions developed in the frame bundle lead to a family of probability distributions on the manifold. We explain how data mean and covariance can be interpreted as points in the frame bundle or, more precisely, in the bundle of symmetric positive definite 2-tensors analogously to the parameters describing Euclidean normal distributions. We discuss a factorization of the frame bundle projection map through this bundle, the natural sub-Riemannian structure of the frame bundle, the effect of holonomy, and the existence of subbundles where the Hormander condition is satisfied such that the Brownian motions have smooth transition densities. We identify the most probable paths for the underlying Euclidean Brownian motion and discuss small time asymptotics of the transition densities on the manifold. The geometric setup yields an intrinsic approach to the estimation of mean and covariance in non-linear spaces.
21 citations
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TL;DR: In this paper, stable holomorphic vector bundles on elliptically fibered Calabi-Yau n-fold Zn were constructed in terms of F-theory compactifications on local singularities.
Abstract: We use a recently proposed construction of stable holomorphic vector bundles V on elliptically fibered Calabi-Yau n-fold Zn in terms of F-theory compactifications on local singularities to describe stability conditions on V. Specifically, the requirement that the F-theory compactification manifold is Calabi-Yau implies a a stability criterion on V which is formulated in terms of the existence of holomorphic sections of certain line bundles.
20 citations