Suppose you have the following setup:

Let $(N,g)$ be a (complete) Riemannian manifold and $M \subset N$ a smooth codimension 1 submanifold, $p \in M$. Then $T_pM$ is a subspace of codimension 1 in $T_pN$. Let $0_p$ be the origin of $T_pN$ and $r$ a positive number small enough such that $exp_p^N: B_r(0_p) \to U_r(p)$ is a diffeomorphism (here $U_r(p)$ denotes the geodesic ball of radius $r$ about $p$).

I want to now look at the local "hyperplane" spanned by geodesics emanating from $p$ and tangent $M$ at $p$, i.e, the image of the map $exp^N_p$ when restricted to $T_pM \cap B_r(0_p)$. This "hyperplane" $S$ is a smooth codimension $1$ submanifold of $M$ with $p \in S$.

When $N = \mathbb R^{n+1}$ and $M=S^n$ (or any codimension 1 manifold, for that matter), $r$ can be taken arbitrarily large, so that one just gets the unique affine hyperplane, meeting $M$ at $p$. Any hyperplane of $\mathbb R^n$ is naturally totally geodesic, so in particular it is a minimal surface, giving rise to a natural question: Is $S$ (locally) always a minimal surface ?

I start by trying to compute the mean curvature of $S$ at $p$, which is easy: Let $X$ be a vector field of $S$ defined in a neighborhood of $p$ with $X(p) \in T_pS = T_pM$. Then, there is a unique $N$-geodesic $\gamma$ with $\dot{\gamma}(p) = X(p)$ and, by definition of $S$, the image of $\gamma$ when restricted to $[0,\epsilon]$ for $\epsilon$ small lies in $S$. This implies that the second fundamental form of $S$ vanishes at $p$, hence in particular that $S$ has zero mean curvature at $p$.

The problem is that at other points $q \in S$, $N$-geodesics emamating at $q$ need not to lie in $S$, not even locally, only those that connect $q$ with $p$. Thus, the second fundamental form does not vanish necessarily, the mean curvature could still be zero, however. Although I am in doubt of this, I haven't come up with an easy illustrating example. Can anybody help me out here?

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    $\begingroup$ I guess the claim is not true. But to give a counterexample is also not easy, as it's true when $N$ has dimension $2$ or $N$ is a space form. $\endgroup$ – user99914 Sep 29 '15 at 11:19
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    $\begingroup$ What exactly is the point of $M$ in your question? Is your question just that "Let $(N,g)$ be a complete Riemannian manifold, let $p\in N$ and let $V$ be a codimension-one subspace of $T_pN$, then is $S = \exp^N_p (V \cap B_r)$ minimal?" $\endgroup$ – Willie Wong Oct 1 '15 at 20:29
  • $\begingroup$ You are right, $M$ itself doesnt serve any important purpose in this question, other than to highlight where the original problem came from ( sphere in R^3). However, there are several follow-up questions involving M, I might pose them sometime later. $\endgroup$ – Berni Waterman Oct 2 '15 at 7:21

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