Let $M^n \to N^{n+1}$ be a Riemannian immersion where $M$ is orientable. Suppose at $p,q \in M$ that there are geodesic balls of $p$ and $q$ which are graphs over the same tangent plane $T_pM$.
Definition of a graph over a tangent plane: If $p \in M$, and $(U,\phi,B(p,r))$ is a chart for $p$ where $B(p,r)$ is a geodesic ball, then the pullback of the connected component of $B(p,r) \cap S$ containing $p$ to $U$ is a Euclidean graph over a Euclidean disk in $T_pM$.
Definition of a normal graph: A normal graph of $u$ over an open set $V \subset M$ is defined to be the set $$ \Gamma_u := \{ \exp_x(u(x)\nu(x)) : x \in V \}, $$ where $\nu$ is the Gauss map for $M$.
Under these conditions, is it true that the geodesic ball around $q$ can be expressed as a normal graph $u$ over a geodesic ball around $p$, where we possibly shrink the radii of the balls? That is to say, two graphs over the same tangent plane are normal graphs over each other?