A geometric space without conjugate points

作者：Bucataru Ioan^*; Dahl Matias F

来源：Balkan Journal of Geometry and Its Applications, 2010, 15(1): 17-40.

• spray space
• geodesic spray
• geodesic variation
• complete lift
• conjugate points
• sub-spray

From a spray space S on a manifold M we construct a new geometric space P of larger dimension with the following properties: (i) geodesics in P are in one-to-one correspondence with parallel Jacobi fields of M; (ii) P is complete if and only if S is complete; (iii) if two geodesics in P meet at one point, the geodesics coincide on their common domain, and P has no conjugate points; (iv) there exists a submersion that maps geodesics in P into geodesics on M.
The 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, the 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.

• 出版日期2010