The geometry of geodesic invariant functions and applications to Landsberg surfaces

Salah G. Elgendi; Zoltan Muzsnay

Home/Archive/arXiv

arXiv · 2408.05848

The geometry of geodesic invariant functions and applications to Landsberg surfaces

Salah G. Elgendi·Zoltan Muzsnay

View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, for a given spray $S$ on an $n$-dimensional manifold $M$, we investigate the geometry of $S$-invariant functions. For an $S$-invariant function $¶$, we associate a vertical subdistribution $\V_¶$ and find the relation between the holonomy distribution and $\V_¶$ by showing that the vertical part of the holonomy distribution is the intersection of \ok{all spaces $\V_{\F_S}$ associated to $\F_S$ where $\F_S$} is the set of all Finsler functions that have the geodesic spray $S$. As an application, we study the Landsberg Finsler surfaces. We prove that a Landsberg surface with $S$-invariant flag curvature is Riemannian or has a vanishing flag curvature. We show that for Landsberg surfaces with non-vanishing flag curvature, the flag curvature is $S$-invariant if and only if it is constant, in this case, the surface is Riemannian. Finally, for a Berwald surface, we prove that the flag curvature is $H$-invariant if and only if it is constant.

Explore related subjects

math.DG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Salah G. Elgendi, Zoltan Muzsnay. 2024-08-11. The geometry of geodesic invariant functions and applications to Landsberg surfaces. https://arxiv.org/abs/2408.05848

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

KEEP EXPLORING

Related papers

arXiv2026

Quantum propagation for Berezin-Toeplitz operators

We describe the asymptotic behaviour of the quantum propagator generated by a Berezin-Toeplitz operator with real-valued principal symbol. We also give precise asymptotics for smoothed spectral projectors associated with the operator in the autonomous case; this leads us to introducting quantum states associated with immersed Lagrangian submanifolds. These descriptions involve geometric quantities of two origins, coming from lifts of the Hamiltonian flow to the prequantum bundle and the canonical bundle respectively. The latter are the main contribution of this article and are connected to the Maslov indices appearing in trace formulas, as will be explained in a forthcoming paper.

math.DG↗

arXiv2026

Classification of compact manifolds with positive isotropic curvature

We show the following result: Let $(M,g_0)$ be a compact manifold of dimension $n\geq 12$ with positive isotropic curvature. Then $M$ is diffeomorphic to a spherical space form, or a quotient manifold of $\mathbb{S}^{n-1}\times \mathbb{R}$ by a cocompact discrete subgroup of the isometry group of the round cylinder $\mathbb{S}^{n-1}\times \mathbb{R}$, or a connected sum of a finite number of such manifolds. This extends previous works of Brendle and Chen-Tang-Zhu, and improves a work of Huang. The proof uses Ricci flow with surgery on compact orbifolds, with the help of the ambient isotopy uniqueness of closed tubular neighborhoods of an isolated singular point in an orbifold.

math.DG↗

arXiv2026

Willmore surfaces in 4-dimensional conformal manifolds

This paper is dedicated to the exploration of the conformal Willmore functional for surfaces within 4-dimensional conformal manifolds. We provide a detailed calculation of both the first and second variations, and present the Euler-Lagrange equation of this functional in a conformally invariant form. Utilizing the second variation formula we derived, we demonstrate that the Clifford torus in $\mathbb{C}P^2$ is strictly Willmore-stable. This finding strongly supports the conjecture proposed by Montiel and Urbano [J. reine angew. Math. 546 2002, 139-154], which posits that the Clifford torus in $\mathbb{C}P^2$ minimizes the Willmore functional among all tori. Moreover, by applying our formula to complex curves in $\mathbb{C}P^2$, we establish that the first nonzero eigenvalue of the Jacobi operator is at least 12. In the context of 4-dimensional locally symmetric spaces, we construct several holomorphic differentials to show that among all minimal 2-spheres, only those super-minimal ones can be Willmore.

math.DG↗