A geometric space without conjugate points

TL;DR

The paper constructs a new geometric space from a spray space that has no conjugate points, preserves geodesic correspondence with Jacobi fields, and maintains completeness, by adding dimensions and relaxing structural assumptions.

Contribution

It introduces a method to create higher-dimensional spaces without conjugate points from spray spaces, expanding geometric possibilities beyond traditional structures.

Findings

Space P has no conjugate points.

Geodesics in P correspond to parallel Jacobi fields.

P is complete if and only if S is complete.

Abstract

From a spray space $S$ on a manifold $M$ we construct a new geometric space $P$ of larger dimension with the following properties: 1. Geodesics in $P$ are in one-to-one correspondence with parallel Jacobi fields of $M$. 2. $P$ is complete if and only if $S$ is complete. 3. If two geodesics in $P$ meet at one point, the geodesics coincide on their common domain, and $P$ has no conjugate points. 4. There exists a submersion $\pi\colon P \to M$ that maps geodesics in $P$ into geodesics on $M$. Space $P$ is constructed by first taking two complete lifts of spray $S$. This will give a spray $S^{cc}$ on the second iterated tangent bundle $TTM$. Then space $P$ is obtained by restricting tangent vectors of geodesics for $S^{cc}$ onto a suitable $(2\dim M+2)$-dimensional submanifold of $TTTM$. Due to the last restriction, space $P$ is not a spray space. However, the construction shows that conjugate points can be removed if we add dimensions and relax assumptions on the geometric structure.