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Showing papers on "Frame bundle published in 2018"


Journal ArticleDOI
TL;DR: A semi-analytic method for matching the boundary conditions is proposed by using the analytic form of the extremal solutions and a closed form solution for the exponential map that has the advantage that an analytic description of the control accelerations can be derived and enables actuator constraints to be incorporated via time reparametrization.

9 citations


Journal ArticleDOI
TL;DR: In this article, the Marsden-Weinstein-Meyer quotient of the manifold was used to describe the relationship between two different parametrisations of Gaussian wave packet dynamics commonly used in semiclassical mechanics.
Abstract: Recently Ohsawa has studied the Marsden-Weinstein-Meyer quotient of the manifold $T^*\mathbb{R}^n\times T^*\mathbb{R}^{2n^2}$ under a $\operatorname{O}(2n)$-symmetry, and has used this quotient to describe the relationship between two different parametrisations of Gaussian wave packet dynamics commonly used in semiclassical mechanics In this paper we suggest a new interpretation of (a subset of) the unreduced space as being the frame bundle $\mathcal{F}(T^*\mathbb{R}^n)$ of $T^*\mathbb{R}^n$ We outline some advantages of this interpretation, and explain how it can be extended to more general symplectic manifolds using the notion of the diagonal lift of a symplectic form due to Cordero and de Le\'on

7 citations


Journal ArticleDOI
TL;DR: In this paper, the authors define Wick-rotations by considering pseudo-Riemannian manifolds as real slices of a holomorphic Riemannians manifold and derive new results regarding the existence, and non-existence, of such Wickrotations.

7 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that an analytic action of a connected Lie group on an analytic manifold M becomes free on a comeager subset of an open subset of M when prolonged to a frame bundle of sufficiently high order.
Abstract: We prove that an effective, analytic action of a connected Lie group G on an analytic manifold M becomes free on a comeager subset of an open subset of M when prolonged to a frame bundle of sufficiently high order. We further prove that the action of G becomes free on a comeager subset of an open subset of a submanifold jet bundle over M of sufficiently high order, thereby establishing a general result that underlies Lie's theory of symmetry groups of differential equations and the equivariant method of moving frames.

6 citations


Journal ArticleDOI
TL;DR: In this article, a necessary and sufficient criterion for a vector bundle E to be a direct image of a line bundle under a surjective etale morphism was given, where the criterion is the existence of a Cartan subalgebra bundle of the endomorphism bundle End(E).

4 citations


Journal ArticleDOI
TL;DR: In this paper, the general form of the Schmidt metric in the case of Lorentzian surfaces is given, and the Ricci scalar of Schmidt metric is defined in terms of the Riemannian scalar.
Abstract: The b-boundary is a mathematical tool used to attach a topological boundary to incomplete Lorentzian manifolds using a Riemaniann metric, called the Schmidt metric, on the frame bundle. In this paper we give the general form of the Schmidt metric in the case of Lorentzian surfaces. Furthermore, we write the Ricci scalar of the Schmidt metric in terms of the Ricci scalar of the Lorentzian manifold and give some examples. Finally, we discuss some applications to general relativity.

2 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that the universal space of holomorphic connections is a torsor, where the pullback of a holomorphic principal bundle to a simple group has a tautological holomorphic connection.
Abstract: Given a holomorphic principal bundle $Q\longrightarrow X$, the universal space of holomorphic connections is a torsor$C_1(Q)$ for $\ad Q\otimes T^*X$ such that the pullback of $Q$ to $C_1(Q)$ has a tautological holomorphic connection. When$X= G/P$, where $P$ is a parabolic subgroup of a complex simple group $G$, and $Q$ is the frame bundle of an ample linebundle, we show that $C_1(Q)$ may be identified with $G/L$, where $L \subset P$ is a Levi factor. We use this identificationto construct the twistor space associated to a natural hyper-K\"ahler metric on $T^*(G/P)$, recovering Biquard's description ofthis twistor space, but employing only finite-dimensional, Lie-theoretic means.

1 citations


Proceedings ArticleDOI
01 Apr 2018
TL;DR: The method is semi-analytic and solves the boundary-value problem arising from the geometric framing of the Pontryagin's maximum principle applied to the vehicle kinematics where the velocities are defined analytically in terms of three parameters.
Abstract: This paper describes a global motion planning method for vehicles with actuator constraints based on the semi-analytic solution of minimum energy-type curves on the frame bundle of connected surfaces of arbitrary cross sectional curvature. The method is semi-analytic and solves the boundary-value problem arising from the geometric framing of the Pontryagin's maximum principle applied to the vehicle kinematics where the velocities are defined analytically in terms of three parameters. Numerical shooting on an iterative Lie group expression of the curve in the group is then employed to match the group boundary conditions. This approach has the advantage that an analytic description of the control accelerations can be derived and enables actuator constraints to be incorporated via time reparametrization. The method is applied to a practical example from space mechanics, the spacecraft docking problem with actuator constraints.

1 citations


Posted Content
TL;DR: In this article, the authors studied necessary and sufficient conditions for the existence of Lorentzian and weak Lorentziian cobordisms between closed smooth manifolds of arbitrary dimension such that the structure group of the frame bundle of the cobordism is Spin(1, n)_0$.
Abstract: We study necessary and sufficient conditions for the existence of Lorentzian and weak Lorentzian cobordisms between closed smooth manifolds of arbitrary dimension such that the structure group of the frame bundle of the cobordism is $\Spin(1, n)_0$. This extends a result of Gibbons-Hawking on $\Sl(2, \C)$-Lorentzian cobordisms between 3-manifolds and results of Reinhart and Sorkin on the existence of Lorentzian cobordisms. We compute the $\Spin(1, n)_0$-Lorentzian cobordism group for several dimensions. Restrictions on the gravitational kink numbers of $\Spin(1, n)_0$-weak Lorentzian cobordisms are obtained.


Journal ArticleDOI
TL;DR: In this paper, the authors proposed a method to preserve translational equivariance / gauge symmetry by defining an affine connection on the affine bundle, defining a zero section on the associated affine vector bundle, and then using this connection and the zero section to define an associated solder form.
Abstract: How to include spacetime translations in fibre bundle gauge theories has been a subject of controversy, because spacetime symmetries are not internal symmetries of the bundle structure group. The standard method for including affine symmetry in differential geometry is to define a Cartan connection on an affine bundle over spacetime. This is equivalent to (1) defining an affine connection on the affine bundle, (2) defining a zero section on the associated affine vector bundle, and (3) using the affine connection and the zero section to define an "associated solder form," whose lift to a tensorial form on the frame bundle becomes the solder form. The zero section reduces the affine bundle to a linear bundle and splits the affine connection into translational and homogeneous parts; however it violates translational equivariance / gauge symmetry. This is the natural geometric framework for Einstein-Cartan theory as an affine theory of gravitation. The last section discusses some alternative approaches that claim to preserve translational gauge symmetry.