Let $M$ be a $n$-dimensional Riemannian Manifold, fix $p\in M$, and $1<k<n$. Do we know if the following is true?

For any $k$-dimensional subspace $V$ of $T_p M$, there exists a minimal submanifold through $p$ and tangential to $V$ at $p$.

Or even better, do we have this?

There exists a diffeomorphism map $F$ from a ball $B\subset T_p M$ centered at 0 to $F(B)\subset M$ such that $F(0)=p$, and for any $k$-dimensional subspace $V$ of $T_p M$, $F(V\cap B)$ is an minimal submanifold.

Of course, if $k=1$, the exponential map will do.

Any comments or references are appreciated.