A Special Nonlinear Connection in Second Order Geometry

Nicoleta Brinzei

Home/Archive/arXiv

arXiv · 0706.4362

A Special Nonlinear Connection in Second Order Geometry

Nicoleta Brinzei

View original source ↗Read source document ↗Export metadata ↓

Abstract

We show that, for mechanical system with external forces, the equations of deviations of solution curves of the corresponding Lagrange equations,determine a nonlinear connection on the second order osculator (second order tangent) bundle. In particular, Jacobi equations in Finsler and Riemann spaces determine such a nonlinear connection.

Explore related subjects

math.DG math-ph math.MP

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Nicoleta Brinzei. 2007-06-29. A Special Nonlinear Connection in Second Order Geometry. https://arxiv.org/abs/0706.4362

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Poisson transforms, the BGG complex, and discrete series representations of SU(n+1,1)

The aim of this article is to construct specific Poisson transforms mapping differential forms on the sphere $S^{2n+1}$ endowed with its natural CR structure to forms on complex hyperbolic space. These transforms have co-closed harmonic values, descend to the BGG (Rumin) complex, and intertwine the differential operators in that complex with the exterior derivative. Passing to the Poincaré ball model, we analyze boundary asymptotics, proving that the values of our transforms admit a continuous extension to the boundary in degrees $\leq n$. Finally, we show that composing the exterior derivative with the transform in degree $n$, one obtains an isomorphism between the kernel of the Rumin operator in degree $n$ and a dense subspace of the $L^2$-harmonic forms on complex hyperbolic space. This provides a realization of the disc rete series representations of $SU(n+1,1)$ with trivial infinitesimal character in spaces of differential forms on the compact manifold $S^{2n+1}$. These developments are motivated by a program of the third author to prove some instances of the Baum-Connes conjecture. The first part of the article is valid in a much more general setting, in particular, it is relevant for cases in which the conjecture is still open.

math.DG↗

arXiv2026

Oriented Grassmannian Bundle, Normal Curvature Reduction, and Persistent Homology

We consider a smooth closed orientable submanifold $M \subset \mathbb{R}^D$ with narrow cycles. We embed $M$ into a scaled oriented Grassmannian bundle via the Gauss map in order to enlarge the scale of these cycles. Under mild assumptions, we show that this embedding reduces the normal curvature of the embedded submanifold in directions where the original normal curvature is large. For smooth closed hypersurfaces, we further show that this construction increases the distance between antipodal points of narrow cycles for fixed volume. We then obtain an explicit range of radii for which the ambient Čech complex on this Grassmannian bundle is homotopy equivalent to the embedded manifold, yielding lower bounds on the scales at which the Čech filtration recovers the homology of $M$. Since the distance induced by the embedding depends on both positions and oriented tangent spaces, we work with Whitney $C^1$ convergence of embeddings and prove that the associated Čech persistent homology is stable with respect to the interleaving distance. Finally, we describe a procedure for computing a distance matrix for a finite subset with respect to this embedding and illustrate the construction on several examples, including an approximate quasi-halo orbit in the Saturn--Enceladus system.

math.DG↗

arXiv2026

Mabuchi solitons and Mabuchi constants on Fano admissible manifolds

In this paper, we study the existence of Mabuchi solitons on admissible manifolds as defined by Apostolov--Calderbank--Gauduchon--Tønnesen-Friedman. We prove that a Fano admissible manifold admits a Mabuchi soliton if and only if the Mabuchi constant is less than 1. We also provide an explicit formula for the Mabuchi constant on Fano admissible manifolds, which generalizes that of Mabuchi. Using this formula, we completely determine the existence and non-existence of Mabuchi solitons on Fano admissible manifolds over the complex projective space $\mathbf{P}^{n}$.

math.DG↗