Skip to content
Free planNo credit card requiredStart writing
AI for Research home
AI for Research home
Dashboard
Sign inGet started

Search 250 million research papers

Read the abstract, see who wrote it and who cites it, and get the citation in APA, MLA, Chicago or Harvard.

About 1,462 papers for “Frame bundle”

Connections and Frame Bundle Reductions

M. Lachièze-Rey · arXiv (Cornell University) · 2020

In general relativity, the gravitational potential is represented by the\nLevi-Civita connection, the only symmetric connection preserving the metric. On\na differentiable manifold, a metric identifies with an orthogonal structure,\ndefined as a Lorentz reduction of the frame bundle. The Levi-Civita connection\nappears as the only symmetric connection preserving the reduction. This paper\npresents generalization of this process to other aproaches of gravitation: Weyl\nstructure with Weyl connections, teleparallel structures with Weitzenbock\nconnections, unimodular structure, similarly appear as frame bundle reductions,\nwith preserving connections. To each subgroup H of the linear group GL\ncorrespond reduced structures, or H-structures. They are subbundles of the\nframe bundle (with GL as principal group), with H as principal group. A linear\nconnection in a manifold M is a principal connection on the frame bundle. Given\na reduction, the corresponding preserving connections on M are the linear\nconnections which preserve it. I also show that the time gauge used in the 3+1\nformalism for general relativity similarly appears as the result of a bundle\nreduction.\n

Cited by 1Open access

Connections and Frame Bundle Reductions

Marc Lachièze-Rey · arXiv (Cornell University) · 2020

In general relativity, the gravitational potential is represented by the Levi-Civita connection, the only symmetric connection preserving the metric. On a differentiable manifold, a metric identifies with an orthogonal structure, defined as a Lorentz reduction of the frame bundle. The Levi-Civita connection appears as the only symmetric connection preserving the reduction. This paper presents generalization of this process to other aproaches of gravitation: Weyl structure with Weyl connections, teleparallel structures with Weitzenbock connections, unimodular structure, similarly appear as frame bundle reductions, with preserving connections. To each subgroup H of the linear group GL correspond reduced structures, or H-structures. They are subbundles of the frame bundle (with GL as principal group), with H as principal group. A linear connection in a manifold M is a principal connection on the frame bundle. Given a reduction, the corresponding preserving connections on M are the linear connections which preserve it. I also show that the time gauge used in the 3+1 formalism for general relativity similarly appears as the result of a bundle reduction.

Cited by 0Open access

Frame bundle approach to generalized minimal submanifolds

Kamil Niedziałomski · arXiv (Cornell University) · 2016

We extend the notion of $r$-minimality of a submanifold in arbitrary codimension to $u$-minimality for a multi-index $u\in\mathbb{N}^q$, where $q$ is the codimension. This approach is based on the analysis on the frame bundle of orthonormal frames of the normal bundle to a submanifold and vector bundles associated with this bundle. The notion of $u$-minimality comes from the variation of $σ_u$-symmetric function obtained from the family of shape operators corresponding to all possible bases of the normal bundle. We obtain the variation field, which gives alternative definition of $u$--minimality. Finally, we give some examples of $u$-minimal submanifolds for some choices of $u$.

Cited by 0Open access

BA-LINS: A Frame-to-Frame Bundle Adjustment for LiDAR-Inertial Navigation

Hailiang Tang, Tisheng Zhang, Liqiang Wang et al. · IEEE Transactions on Intelligent Transportation Systems · 2025

Bundle adjustment (BA) has been proven to improve the accuracy of the LiDAR mapping, but has not yet been properly employed in a dead-reckoning navigation system. In this paper, we present a frame-to-frame (F2F) BA for LiDAR-inertial navigation, named BA-LINS. Based on the direct F2F point-cloud association method, the same-plane points are associated among the LiDAR keyframes. Hence, the F2F plane-point BA measurement model can be constructed using the same-plane points. The LiDAR BA and the inertial measurement unit (IMU)-preintegration measurements are tightly coupled under the framework of factor graph optimization. Meanwhile, an effective adaptive covariance estimation algorithm for LiDAR BA measurements is proposed to improve the accuracy further. Exhaustive experiment results on public and private datasets demonstrate that BA-LINS yields superior accuracy to state-of-the-art methods. Compared to the baseline system FF-LINS, the absolute translation accuracy and state-estimation efficiency of BA-LINS are improved by 29.5% and 28.7%, respectively. Besides, the proposed adaptive covariance estimation algorithm exhibits notably improved accuracy and robustness.

Cited by 3

Principal bundle structures among second order frame bundles

M. de León, Abel Méndez · arXiv (Cornell University) · 2015

Using a model for the bundle $\hat{\mathcal F}^2M$ of semi-holonomic second order frames of a manifold $M$ as an extension of the bundle ${\mathcal F}^2M$ of holonomic second order frames of $M$, we introduce in $\hat{\mathcal F}^2M$ a principal bundle structure over ${\mathcal F}^2M$, the structure group being the additive group $A_2(n)$ of skew-symmetric bilinear maps from ${\Bbb R}^2 \times {\Bbb R}^n$ into ${\Bbb R}^n$. The composition of the projection of that structure with the existing projection of the bundle $\tilde{\mathcal F}^2M$ of non-holonomic second order frames of $M$ over $\hat{\mathcal F}^2M$ provides a principal bundle structure in $\tilde{\mathcal F}^2M$ over ${\mathcal F}^2M$. These results close an existing gap in the theory of second order frame bundles.

Cited by 0Open access

The Geometry of the Frame Bundle over Spacetime

2008

One of the known mathematical descriptions of singularities in General Relativity is the b-boundary, which is a way of attaching endpoints to inextendible endless curves in a spacetime. The b-boundary of a manifold M with connection Γ is constructed by forming the Cauchy completion of the frame bundle LM equipped with a certain Riemannian metric, the b-metric G. We study the geometry of (LM, G) as a Riemannian manifold in the case when Γ is the Levi-Cività connection of a Lorentzian metric g on M. In particular, we give expressions for the curvature and discuss the isometries and the geodesics of (LM, G) in relation to the geometry of (M, g). 1

Cited by 0

The Geometry of the Frame Bundle over Spacetime

Staahl, F · CERN Bulletin · 2000

One of the known mathematical descriptions of singularities in General Relativity is the b-boundary, which is a way of attaching endpoints to inextendible endless curves in a spacetime. The b-boundary of a manifold M with connection is constructed by forming the Cauchy completion of the frame bundle LM equipped with a certain Riemannian metric, the b-metric G. We study the geometry of (LM,G) as a Riemannian manifold in the case when the connection is the Levi-Civita connection of a Lorentzian metric g on M. In particular, we give expressions for the curvature and discuss the isometries and the geodesics of (LM,G) in relation to the geometry of (M,g).

Cited by 0

A Frame Bundle Generalization of Multisymplectic Momentum Mappings

Lawson, J. K. · arXiv (Cornell University) · 2001

This paper presents generalized momentum mappings for covariant Hamiltonian field theories. The new momentum mappings arise from a generalization of symplectic geometry to $L_VY$, the bundle of vertically adapted linear frames over the bundle of field configurations $Y$. Specifically, the generalized field momentum observables are vector-valued momentum mappings on the vertically adapted frame bundle generated from automorphisms of $Y$. The generalized symplectic geometry on $L_VY$ is a covering theory for multisymplectic geometry on the multiphase space $Z$, and it follows that the field momentum observables on $Z$ are generalized by those on $L_VY$. Furthermore, momentum observables on $L_VY$ produce conserved quantities along flows in $L_VY$. For translational and orthogonal symmetries of fields and reparametrization symmetry in mechanics, momentum is conserved, and for angular momentum in time-evolution mechanics we produce a version of the parallel axis theorem of rotational dynamics, and in special relativity, we produce the transformation of angular momentum under boosts.

Cited by 0Open access

On the frame bundle adapted to a submanifold

Kamil Niedziałomski · Mathematische Nachrichten · 2014

Let M be a submanifold of a Riemannian manifold . M induces a subbundle of adapted frames over M of the bundle of orthonormal frames . The Riemannian metric g induces a natural metric on . We study the geometry of a submanifold in . We characterize the horizontal distribution of and state its correspondence with the horizontal lift in induced by the Levi–Civita connection on N. In the case of extrinsic geometry, we show that minimality is equivalent to harmonicity of the Gauss map of the submanifold M with a deformed Riemannian metric. In the case of intrinsic geometry we compute the curvatures and compare this geometry with the geometry of M.

Cited by 4

Principal bundle structure on jet prolongations of frame bundles

Ján Brajerčík, Milan Demko, Demeter Krupka · Mathematica Slovaca · 2014

Abstract In this paper, we introduce the structure of a principal bundle on the r-jet prolongation J r FX of the frame bundle FX over a manifold X. Our construction reduces the well-known principal prolongation W r FX of FX with structure group G nr. For a structure group of J r FX we find a suitable subgroup of G nr. We also discuss the structure of the associated bundles. We show that the associated action of the structure group of J r FX corresponds with the standard actions of differential groups on tensor spaces.

Cited by 1Open access

Conceptual frame work on bundle care therapy on hypertension

Katari Kantha, J Jasmine, Manjubala Dash et al. · International Journal of Applied Research · 2022

Present health studies show that the world is heading for heart diseases of worldwide segment. The number of people at high risk from cardiovascular disease is increasing recent studies suggest that only 2 to 7 percent of the common population have no risk factors at all, and less than 70 percent individuals have multiple risk factors. Non-communicable diseases (NCDs) are a cluster of diseases with long duration, and are generally slow in progression that affect individuals over an extended period of time causing socio economic burden to the nation. The good result of a bundle comes from the body of science behind it and the way of execution. A bundle bounds the changes together into a package of involvement which people know must be followed for every patient, every single time. Bundles may resemble a checklist but it is more than that. Bundles are different from a checklist as the latter contains many elements (more than five usually), all are not necessarily evidence based. The hypertensive safety bundle care therapy (HSBCT) on cardiovascular disease was followed. HSBCT includes Diet - DASH diet with low carbohydrate, salt restricted and high fiber diet, physical activity, spiritual practices by prayer, stress management with Pranayama, reduction of Habits like alcohol and smoking, Obesity management regular intake of Medications and regular Follow up. Thus, a multimodal care so called hypertensive safety bundle care therapy on prevention of risk for CAD (cardiovascular disease) among hypertensive patients may yield a good result by reducing the CAD risk factors.

Cited by 0Open access