In this note, we prove the Morse index theorem for a geodesic connecting two submanifolds in a $C^7$ manifold with a $C^6$ (conic) pseudo-Finsler metric provided that the fundamental tensor is positive definite along velocity curve of the geodesic.
In this paper, we introduce isoparametric functions and isoparametric hypersurfaces in conic Finsler spaces. We find that in a conic Minkowski space, besides the conic Minkowski hyperplanes, conic Minkowski hyperspheres and conic Minkowski cylinders, which are all isoparametric hypersurfaces, there are probably other isoparametric hypersurfaces, such as helicoids. Moreover, we give a complete classification of isoparametric hypersurfaces in kropina spaces with constant flag curvature.
In this paper, we study left invariant conic Finsler metrics on the 2-dimensional non-Abelian Lie group $G$ with nowhere vanishing spray vector fields, and classify those satisfying the constant curvature condition, the Landsberg condition or the Berwald condition respectively. We prove that any left invariant conic Landsberg metric on $G$ must be Berwald. This discovery enable us to propose a homogeneous conic Landsberg Conjecture, which guesses that every homogeneous conic Landsberg metric is Berwald, and prove the 2-dimensional case for it.
We give the details of the proof of the equality between the critical groups, with respect the H^1 and C^1 topology, at a non-degenerate critical point of the energy functional of a non-reversible Finsler manifold (M,F), defined on the Hilbert manifold of the H^1 curves connecting two given points on M.
We show that the index of a lightlike geodesic in a conformally standard stationary spacetime is equal to the index of its spatial projection as a geodesic of a Finsler metric associated to the spacetime. Moreover we obtain the Morse relations of lightlike geodesics connecting a point to an integral line of the standard timelike Killing vector field by using Morse theory on the associated Finsler manifold. To this end, we prove a splitting lemma for the energy functional of a Finsler metric. Finally, we show that the reduction to Morse theory of a Finsler manifold can be done also for timelike…
For a standard Finsler metric F on a manifold M, its domain is the whole tangent bundle TM and its fundamental tensor g is positive-definite. However, in many cases (for example, the well-known Kropina and Matsumoto metrics), these two conditions are relaxed, obtaining then either a pseudo-Finsler metric (with arbitrary g) or a conic Finsler metric (with domain a "conic" open domain of TM). Our aim is twofold. First, to give an account of quite a few subtleties which appear under such generalizations, say, for conic pseudo-Finsler metrics (including, as a previous step, the case of Minkowski c…
We prove the shifting theorems of the critical groups of critical points and critical orbits for the energy functionals of Finsler metrics on Hilbert manifolds of $H^1$-curves, and two splitting lemmas for the functionals on Banach manifolds of $C^1$-curves. Two results on critical groups of iterated closed geodesics are also proved; their corresponding versions on Riemannian manifolds are based on the usual splitting lemma by Gromoll and Meyer (1969). Our approach consists in deforming the square of the Finsler metric in a Lagrangian which is smooth also on the zero section and then in using …
The goal of this article is to generalise the Witten deformation to even dimensional conic manifolds and a class of functions called admissible Morse functions.
We show that any two non-conjugate points on a forward or backward complete connected Finsler manifold can be joined by infinitely many geodesics which are not covered by finitely many closed ones, provided that the Betti numbers of the based loop space grow unbounded.
In the present paper, we introduce and investigate the notion of a semi concurrent vector field on a Finsler manifold. We show that some special Finsler manifolds admitting such vector fields turn out to be Riemannian. We prove that Tachibana's characterization of Finsler manifolds admitting a concurrent vector field leads to Riemannain metrics. We give an answer to the question raised in \cite{DWF}: "Is any n-dimensional Finsler manifold $(M,F)$, admitting a non-constant smooth function $f$ on $M$ such that $\frac{\partial f}{\partial x^i}\frac{\partial g^{ij}}{\partial y^k}=0$, a Riemannian …
In this paper, we present a sufficient condition for finite Morse index of complete properly self-shrinkers. We prove that a complete properly embedded self-shrinker in $\mathbb{R}^{n+1}$ with finite asymptotically conical ends or asymptotically cylindrical ends must have finite Morse index. Moreover, as an application of this result, we show that a complete properly embedded self-shrinker in $\mathbb{R}^3$ with finite genus has finite Morse index.
In this work, a version of Fermat's principle for causal curves with the same energy in time orientable Finsler spacetimes is proved. We calculate the secondvariation of the {\it time arrival functional} along a geodesic in terms of the index form associated with the Finsler spacetime Lagrangian. Then the character of the critical points of the time arrival functional is investigated and a Morse index theorem in the context of Finsler spacetime is presented.
This is a survey paper on Morse theory and the existence problem for closed geodesics. The free loop space plays a central role, since closed geodesics are critical points of the energy functional. As such, they can be analyzed through variational methods. The topics that we discuss include: Riemannian background, the Lyusternik-Fet theorem, the Lyusternik-Schnirelmann principle of subordinated classes, the Gromoll-Meyer theorem, Bott's iteration of the index formulas, homological computations using Morse theory, $SO(2)$- vs. $O(2)$-symmetries, Katok's examples and Finsler metrics, relations t…
In some recent papers, the relations existing between the metric properties of Randers spaces and the conformal geometry of stationary Lorentzian manifolds were discovered and investigated. In this note, we focus on the equality between the index of a geodesic in a Randers space and that of its lightlike lift in the associated conformal stationary spacetime. Moreover we make some remarks about regularity of the energy functional of a Finsler metric on the infinite dimensional manifold of $H^1$ curves between two points, in connection with infinite dimensional techniques in Morse Theory.
If all prime closed geodesics on $(S^n,F)$ with an irreversible Finsler metric $F$ are irrationally elliptic, there exist either exactly $2\left[\frac{n+1}{2}\right]$ or infinitely many distinct closed geodesics. As an application, we show the existence of three distinct closed geodesics on bumpy Finsler $(S^3,F)$ if any prime closed geodesic has non-zero Morse index.
We establish the splitting lemmas (or generalized Morse lemmas) for the energy functionals of Finsler metrics on the natural Hilbert manifolds of $H^1$-curves around a critical point or a critical $\R^1$ orbit of a Finsler isometry invariant closed geodesic. They are the desired generalization on Finsler manifolds of the corresponding Gromoll-Meyer's splitting lemmas on Riemannian manifolds (\cite{GM1, GM2}). As an application we extend to Finsler manifolds a result by Grove and Tanaka \cite{GroTa78, Tan82} about the existence of infinitely many, geometrically distinct, isometry invariant clos…
It is well known that a system of homogeneous second-order ordinary differential equations (spray) is necessarily isotropic in order to be metrizable by a Finsler function of scalar flag curvature. In Theorem 3.1 we show that the isotropy condition, together with three other conditions on the Jacobi endomorphism, characterize sprays that are metrizable by Finsler functions of scalar flag curvature. The proof of Theorem 3.1 provides an algorithm to construct the Finsler function of scalar flag curvature, in the case when a given spray is metrizable. One condition of Theorem 3.1, regarding the r…
We show that for an open and dense set non-reversible Finsler metrics on a sphere of odd dimension $n=2m-1 \ge 3$ there is a second closed geodesic with Morse index $\le 4(m+2)(m-1)+2.$
A short survey on the type numbers of closed geodesics, on applications of the Morse theory to proving the existence of closed geodesics and on the recent progress in applying variational methods to the periodic problem for Finsler and magnetic geodesics
In this paper, we first generalize the common index jump theorem for symplectic matrix paths proved in 2002 by Long and Zhu in [LoZ], and get an enhanced version of it. As its applications, we further prove that for a compact simply-connected manifold $(M,F)$ with a bumpy, irreversible Finsler metric $F$ and $H^*(M;{\bf Q})\cong T_{d,n+1}(x)$ for some even integer $d\ge 2$ and integer $n\ge 1$, there exist at least $\frac{dn(n+1)}{2}$ distinct non-hyperbolic closed geodesics with odd Morse indices, provided the number of distinct prime closed geodesics is finite and every prime closed geodesic…
Recently, F. Balacheff proved that the Calabi-Croke sphere made of two flat 1-unit-side equilateral triangles glued along their boundaries is a local extremum for the length of the shortest closed geodesic among the Riemannian spheres with conical singularities of fixed area. We give an alternative proof of this theorem, which does not make use of the uniformization theorem, and extend the result to Finsler metrics.
We consider any Finsler metric on a closed, orientable surface of genus greater than one. H. M. Morse proved that we can associate an asymptotic direction to minimal rays in the universal cover (in the Poincaré disc: a point on the unit circle). We prove here that, if two minimal rays have a common asymptotic direction, which is not a fixed point of the group of deck transformations, then the two rays can intersect at most in a common initial point. This has strong consequences for the structure of the set of minimal geodesics, as well as for the set of Busemann functions associated to the Fin…
We consider a Morse function $f$ and a Morse-Smale gradient-like vector field $X$ on a compact connected oriented 3-manifold $M$ such that $f$ has only one critical point of index 3. Based on Laudenbach's ideas, we will show that the flow of $X$ can be isotoped into one so that the trajectory spaces of the new flow provide a stratification for $M$. We will construct "natural" tubular neighborhoods about each given trajectory space of the new flow such that these neighborhoods are stratified by open subsets of trajectory spaces that co-bound the given one. In connection with this we introduce t…
