Connections on Tangent Bundles of Higher Order Associated to Regular Lagrangians (original) (raw)

On Nonlinear Connections in Higher Order Lagrange Spaces

Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis, 2008

Considering a Lagrangian of order k, we determine a nonlinear connection N on T k M such that the horizontal and vertical distributions to be Lagrangian subbundles for the presymplectic structure given by the Cartan-Poincaré two-form ω k L .

On the Optimal Regularity Implied by the Assumptions of Geometry I: Connections on Tangent Bundles

arXiv (Cornell University), 2019

We resolve the problem of optimal regularity and Uhlenbeck compactness for affine connections in General Relativity and Mathematical Physics. First, we prove that any affine connection Γ, with components Γ ∈ L 2p and components of its Riemann curvature Riem(Γ) in L p , in some coordinate system, can be smoothed by coordinate transformation to optimal regularity, Γ ∈ W 1,p (one derivative smoother than the curvature), p > max{n/2, 2}, dimension n ≥ 2. For Lorentzian metrics in General Relativity this implies that shock wave solutions of the Einstein-Euler equations are non-singular-geodesic curves, locally inertial coordinates and the Newtonian limit, all exist in a classical sense, and the Einstein equations hold in the strong sense. The proof is based on an L p existence theory for the Regularity Transformation (RT) equations, a system of elliptic partial differential equations (introduced by the authors) which determine the Jacobians of the regularizing coordinate transformations. Secondly, this existence theory gives the first extension of Uhlenbeck compactness from Riemannian metrics, to general affine connections bounded in L ∞ , with curvature in L p , p > n, including semi-Riemannian metrics, and Lorentzian metric connections of relativistic Physics. We interpret this as a "geometric" improvement of the generalized Div-Curl Lemma. Our theory shows that Uhlenbeck compactness and optimal regularity are pure logical consequences of the rule which defines how connections transform from one coordinate system to another-what one could take to be the "starting assumption of geometry". M. REINTJES AND B. TEMPLE 11. Proof of Lemmas 10.6, 10.7, 10.9 and 10.10 68 12. Extension of the existence theory to L p connections 78 13. Proof of Theorems 3.1 and 3.2 82 Appendix A. Sobolev norms and inequalities 87 Appendix B. Elliptic PDE theory 88 Appendix C. Cauchy Riemann type equations at low regularities 90 Declarations and Statements 97 Funding 98 Acknowledgements 98 References 98

A Lagrangian form of tangent forms

Journal of Geometry and Physics, 2014

The aim of the paper is to study some dynamic aspects coming from a tangent form, i.e. a time dependent differential form on a tangent bundle. The action on curves of a tangent form is natural associated with that of a second order Lagrangian linear in accelerations, while the converse association is not unique. An equivalence relation of tangent form, compatible with gauge equivalent Lagrangians, is considered. We express the Euler-Lagrange equation of the Lagrangian as a second order Lagrange derivative of a tangent form, considering controlled and higher order tangent forms. Hamiltonian forms of the dynamics generated are given, extending some quantization formulas given by Lukierski, Stichel and Zakrzewski. Using semi-sprays, local solutions of the E-L equations are given in some special particular cases.

On the regularity implied by the assumptions of geometry II -- Connections on vector bundles

arXiv (Cornell University), 2021

We establish optimal regularity and Uhlenbeck compactness for connections on vector bundles over arbitrary base manifolds, including both Riemannian and Lorentzian manifolds, and allowing for both compact and noncompact Lie groups. The proof is based on the discovery of a non-linear system of elliptic equations, (the RT-equations), which provides the coordinate and gauge transformations that lift the regularity of a connection to one derivative above its L p curvature. This is a new mathematical principle. It extends theorems of Kazdan-DeTurck and Uhlenbeck from Riemannian to the Lorentzian geometry of General Relativity, including Yang-Mills connections of relativistic Physics. CONTENTS 1. Introduction 1 2. Statement of Results 4 3. Preliminaries 10 4. The RT-equations for vector bundles 12 5. Existence theory for the RT-equations-Proof of Theorem 2.2 18 6. Weak formalism 27 7. Proof of optimal regularity and Uhlenbeck compactness 30 Appendix A. Basic results from elliptic PDE theory 33 References 34

Lagrangians and higher order tangent spaces

Balkan Journal of Geometry and Its Applications

The aim of the paper is to prove that (TM)-M-k, the tangent space of order k >= 1 of a manifold M, is diffeomorphic with (TkM)-M-1, the tangent space of k(1)-velocities, and also with (T-k(1))* M, the cotangent space of k(1)-covelocities, via suitable Lagrangians. One prove also that a hyperregular Lagrangian of first order on M can give rise to such diffeomorphisms.

Canonical Torsion-Free Connections on the Total Space of the Tangent and the Cotangent Bundle

In this paper we define a class of torsion-free connections on the total space of the (co-)tangent bundle over a base-manifold with a connection and for which tangent spaces to the fibers are parallel. Each tangent space to a fiber is flat for these connections and the canonical projection from the (co-)tangent bundle to the base manifold is totally geodesic. In particular cases the connection is metric with signature (n,n) or symplectic and admits a single parallel totally isotropic tangent n-plane.

On metric connections with torsion on the cotangent bundle with modified Riemannian extension

Journal of Geometry, 2018

Let M be an n−dimensional differentiable manifold equipped with a torsion-free linear connection ∇ and T * M its cotangent bundle. The present paper aims to study a metric connection ∇ with nonvanishing torsion on T * M with modified Riemannian extension g ∇,c. First, we give a characterization of fibre-preserving projective vector fields on (T * M, g ∇,c) with respect to the metric connection ∇. Secondly, we study conditions for (T * M, g ∇,c) to be semi-symmetric, Ricci semi-symmetric, Z semi-symmetric or locally conharmonically flat with respect to the metric connection ∇. Finally, we present some results concerning the Schouten-Van Kampen connection associated to the Levi-Civita connection ∇ of the modified Riemannian extension g ∇,c. Mathematics subject classification 2010. 53C07, 53C35, 53A45.

Equivalent definitions for connections in higher order tangent bundles

2018

We establish a one to one correspondence between the\linebreak connections C(k−1)C^{(k-1)}C(k−1) (in the bundle % T^{k}M\rightarrow M, used by R. Miron in his work on higher order spaces) and C(0)C^{(0)}C(0) (in the affine bundle % T^{k}M\rightarrow T^{k-1}M, used for example in \cite{CSC1}).