The Cartan group is the free nilpotent Lie group of rank 2 and step 3. We consider the left-invariant sub-Riemannian problem on the Cartan group defined by an inner product in the first layer of its Lie algebra. This problem gives a nilpotent approximation of an arbitrary sub-Riemannian problem with the growth vector (2,3,5). In previous works, we described a group of symmetries of the sub-Riemannian problem on the Cartan group, and the corresponding Maxwell time — the first time when symmetric geodesics intersect one another. It is known that geodesics are not globally optimal after the Maxwell time. In this work, we study local optimality of geodesics on the Cartan group. We prove that the first conjugate time along a geodesic is not less than the Maxwell time corresponding to the group of symmetries. We characterize geodesics for which the first conjugate time is equal to the first Maxwell time. Moreover, we describe continuity of the first conjugate time near infinite values.

Cartan 群是阶 2 和步骤 3 的自由幂零李群。我们考虑由其李代数第一层的内积定义的 Cartan 群上的左不变子黎曼问题。该问题给出了具有增长向量 (2,3,5) 的任意子黎曼问题的幂零近似。在之前的工作中，我们描述了嘉当群上的子黎曼问题的一组对称性，以及相应的麦克斯韦时间——对称测地线第一次相互交叉。众所周知，在麦克斯韦时间之后，测地线不是全局最优的。在这项工作中，我们研究了 Cartan 群上测地线的局部最优性。我们证明沿测地线的第一共轭时间不小于对应于对称群的麦克斯韦时间。我们描述了第一共轭时间等于第一麦克斯韦时间的测地线。此外，我们描述了接近无穷大值的第一共轭时间的连续性。