Construct a hypersurface with fixed principal curvatures at a point

Question

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.

Answer 1

Think that you have two Riemannian metrics in a neighborhood of the origin in $\mathrm{T}_p$, the first is standard Euclidean and the second is induced by $\mathrm{exp}_p$ from $M$.

These two metrics coincide at the origin up to first order. It is sufficient to conclude that principle curvatures at the origin calculated in both metrics are the same.

Answer 2

Eschenburg's example is overkill. Here is a less elegant version of Anton's answer.

First, choose local coordinates $x=(x^1, \dots, x^n)$ near $p$ so that $x(p) = 0$, $g_{ij}(p) = \delta_{ij}$ and $\partial_kg_{ij}(p) = 0$. Using exponential coordinates for this is overkill. It's easy to prove this directly.

Give a unit $N \in T_pM$, the coordinates can be chosen so that at $p$, $N = \partial_n$.

If $$ S = \{ x^n = f(x^1, \dots, x^{n-1}) \}, $$ where $f(0) = 0$ and $\partial_kf(0)=0$, $1 \le k \le n-1$, then you can show that the second fundamental form of $S$ at $p$ is the Hessian of $f$, $$ \nabla^2_{ij}f(p) = \partial^2_{ij}f(0) + \Gamma^k_{ij}\partial_kf(0). $$

Here, $\Gamma^k_{ij}(0) = 0$ and therefore it suffices to let $$ f(x^1, \dots, x^{n-1}) = \frac{a}{2}((x^1)^2+\cdots (x^{n-1})^2. $$