I'm reading Eschenburg's paper Local convexity and nonnegative curvature - Gromov's proof of the sphere theorem recently. And I meet a little question: Given a point $p\in M, N\in T_pM$, we want to construct a hypersurface $S$ around $p$ such that $D_XN=aX$ for all $X\in T_pS$, where $a>0$. He gives the construction as follows: $$ S=exp_p(\partial B_{-\frac{1}{a}N}(\frac{1}{a})\cap V). $$ where $V$ is a neighborhood such that $exp|V$ is a diffeomorphism. But I can't verify this conclusion.

I'm reading Eschenburg's paper Local convexity and nonnegative curvature — Gromov's proof of the sphere theorem recently. And I meet a little question: Given a point $p\in M$, $N\in T_pM$, we want to construct a hypersurface $S$ around $p$ such that $D_XN=aX$ for all $X\in T_pS$, where $a>0$. He gives the construction as follows: $$ S=\exp_p(\partial B_{-\frac{1}{a}N}(\frac{1}{a})\cap V), $$ where $V$ is a neighborhood such that $\exp\rvert V$ is a diffeomorphism. But I can't verify this conclusion.