Let $M$ be a complete Riemannian manifold all of whose sectional curvature are non-positive. Let $F$ be a submanifold of $M$ isometric to $\mathbf R^k$ passing through a point $p$ of $M$.
Then $F$ is the image of a $k$-dimensional linear subspace of $T_pM$ under the geodesic exponential $\exp_p:T_pM\to M$.
I am unable to prove this. What is know is that by the Cartan-Hadamard Theorem $\exp_p:T_pM\to M$ is a covering map.
EDIT The statement which made me ask the question is quoted below.
In any complete Riemannian manifold $M$ of non-positive curvature, if $F\subset M$ contains $p$ and is isometric to $\mathbf R^k$ (the book uses the notation $\mathbf E^k$ instead of $\mathbf R^k$), then $F$ is the image under the exponential map of a $k$-dimensional subspace in $T_pM$.
The above is the first line of the proof of Proposition 10.45 in the image attached below. Here $P(n, \mathbf R)$ is the set of all the $n\times n$ positive definite real matrics and $S(n, \mathbf R)$ is the set of all the $n\times n$ real symmetric matrices.
The text is Bridson and Haefliger's Metric Spaces on Non-Positive Curvature.