Time-optimal solutions of Parallel navigation and Finsler geodesics

A geometric approach to kinematics in control theory is illustrated. A non-linear control system is derived for the problem and the Pontryagin maximum principle is used to find the time-optimal trajectories of the Parallel navigation. It is proved that the time-optimal relative trajectories of the Parallel navigation are geodesics of a Finsler metric. It is notable that the approach has the advantages using feedback.     

Holomorphic curvature of Finsler metrics and complex geodesics

In his famous 1981 paper, Lempert proved that given a point in a strongly convex domain the complex geodesics (i.e., the extremal disks) for the Kobayashi metric passing through that point provide a very useful fibration of the domain. In this paper we address the question whether, given a smooth complex Finsler metric on a complex manifoldM, it is possible to find purely differential geometric properties of the metric ensuring the existence of such a fibration in complex geodesies ofM. We first discuss at some length the notion of holomorphic sectional curvature for a complex Finsler metric; then, using the differential equation of complex geodesies we obtained in [AP], we show that for every pair (p;v) ∈T M, withv ≠ 0, there is a (only a segment if the metric is not complete) complex geodesic passing throughp tangent tov iff the Finsler metric is Kähler, has constant holomorphic sectional curvature ?4, and its curvature tensor satisfies a specific simmetry condition—which are the differential geometric conditions we were after. Finally, we show that a complex Finsler metric of constant holomorphic sectional curvature ?4 satisfying the given simmetry condition on the curvature is necessarily the Kobayashi metric.     

Multiplicity of closed geodesics on Finsler spheres with irrationally elliptic closed geodesics

If all prime closed geodesics on (Sⁿ, 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 (S³, F) if any prime closed geodesic has non-zero Morse index.     

Stability of closed geodesics on Finsler 2-spheres

In this paper, we prove that on every Finsler 2-dimensional sphere, either there exist infinitely many prime closed geodesics or there exist at least two irrationally elliptic prime closed geodesics.     

Weierstrass points and multiple intersection points of closed geodesics

We classify smooth complex projective surfaces of degreed and class , satisfying either (i) –d16, or (ii) 25. All these surfaces are rational or ruled. Indeed, we prove that the smallest value of the class of a non-ruled surface is 30 and in fact there are at least two surfacesS, both of degreed=10 and sectional genusg=6, with Kodaira dimension (S)=0 and class =30. Finally, we classify the smoothk-folds (k3) whose sectional surface has class 23.     

Multiplicity and stability of closed geodesics on bumpy Finsler 3-spheres

We prove that for every Q-homological Finsler 3-sphere (M, F) with a bumpy and irreversible metric F, either there exist two non-hyperbolic prime closed geodesics, or there exist at least three prime closed geodesics. Huagui Duan: Partially supported by NNSF and RFDP of MOE of China. Yiming Long: Partially supported by the 973 Program of MOST, Yangzi River Professorship, NNSF, MCME, RFDP, LPMC of MOE of China, S. S. Chern Foundation, and Nankai University.     

Fixed points and uniqueness of complex geodesics

In this work we examine the conditions which guarantee the uniqueness of a complex geodesic whose range contains two fixed points of a holomorphic mapf of a bounded convex circular domain in itself and is contained in the fixed points set off.     

On the average indices of closed geodesics on positively curved Finsler spheres

In this paper, we prove that on every Finsler n-sphere (S ⁿ , F) for n ≥  6 with reversibility λ and flag curvature K satisfying ${(\frac{\lambda}{\lambda+1})^2 \, < \, K \, \le \, 1}$ , either there exist infinitely many prime closed geodesics or there exist ${[\frac{n}{2}]-2}$ closed geodesics possessing irrational average indices. If in addition the metric is bumpy, then there exist n?3 closed geodesics possessing irrational average indices provided the number of prime closed geodesics is finite.     

The existence of two closed geodesics on every Finsler 2-sphere

In this paper, we prove that for every Finsler metric on S ² there exist at least two distinct prime closed geodesics.     

Covering numbers of manifolds and critical points of a morse function

For a closed connected triangulatedn-manifoldM, we study some numerical invariants (namedcategory andcovering numbers) ofM which are strictly related to the topological structure ofM. We complete the classical results of 3-manifold topology and then we prove some characterization theorems in higher dimensions. Finally some applications are given about the minimal number of critical points (resp. values) of Morse functions defined on a closed connected smoothn-manifold. Work performed under the auspices of the G.N.S.A.G.A. of the C.N.R. and financially supported by the M.P.I. of Italy within the project “Geometria delle Varietà Differenziabili”.     

Trajectory nets connecting all critical points of a smooth function

The disconnected components of certain trajectory nets containing all critical points of a differentiable functionf can be connected by suitably chosen contour sets of a certain associated functiong. A recursive construction yields a locally 1-dimensional connected set Ω which contains all critical points. The possible ways of tracing this set numerically are discussed. This work was supported by the Deutsche Forschungsgemeinschaft.     

Finsler manifolds without conjugate points and with integral Ricci curvature

We prove that the integral of the Ricci curvature on the unit tangent bundle SM of a complete Finsler manifold M without conjugate points is nonpositive and vanishes only if M is flat, provided that the Ricci curvature on SM has an integrable positive or negative part.     

Conjugate points of the causal geodesics of Gödel's cosmological model

Translated from Ukrainskii Geometricheskii Sbornik, No. 31, pp. 47–56, 1988.