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Must a surface obtained by exponentiating a plane in a tangent space of a Riemannian manifold be geodesically convex?

The one dimensional geodesic submanifolds of a given Riemannian manifold $(M,g)$ are just geodesics. So one can can wonder, how to construct 2-dimensional geodesic submanifolds. Lets first consider the following question:

Given any point $x\in M$ and a two dimensional subspace of $V\subset T_xM$. Then the exponential map restricted to a sufficiently small ball around $0\in V$ gives an embedding of the open disc into $M$. When is it a geodesic submanifold?

Note that there are many spaces, that have this property at every point and at every tangent plane, like $\mathbb{S}^n,\mathbb{H}^n,\mathbb{R}^n$ and (if I am not mistaken) products of those. So one can also ask:

What properties must the metric $g$ have to ensure, that at every point $x\in M$ and at every two dimensional subspace $V\subset T_xM$ the exponential map $B_\varepsilon(0)\subset V\rightarrow M$ gives locally geodesic surfaces?

A "general" manifold should not have this property I think. It would be nice to have a simple and short counterexample.