Properties of Totally geodesic submanifolds

Suppose I have a totally geodesic manifold $M$ in a riemannian manifold $(N,g)$. Let $p\in M$ and consider the set $\exp_{p}^{-1}(M)$ where this exponential map is defined in $N$. Will we have that $\exp_{p}^{-1}(M)$ is a linear subspace of $T_pN$? That is if we take $y,z\in \exp_{p}^{-1}(M)$ then $\exp_p(y+z)\in M$?

If not, what if we impose more conditions,i.e. that we have a riemannian submersion $\pi:N\rightarrow M$, can we get a better result here?

This question came up cause I am trying to model a Banach manifold of curves using the exponential map and they need to satisfy a boundary condition with respect to a submanifold.

Edit: What about if we restrict to a neighborhood where $\exp_p$ is a diffeomorphism. Say there exists an $r>0$ such that $\exp_p|_{B_{r}(0)}$ is a diffeomorphism. Therefore, since a geodesic in $M$ lifts to geodesic in $N$ we obtain that $\exp_p^{-1}(M)\subset T_pM$. Then we have $y+z\in T_pM$ and so $\exp(y+z)\in M$ due to the uniquess of geodesics and the fact that a geodesic in $M$ lifts to a geodesic in $N$.

Any help and comments is appreciated, thanks in advance.

Certainly, $exp _p (T_pM ) \subset M$ as any geodesic issued from $p$ and tangent to $M$ stays in $M$. And certainly $exp_p^{-1}(M)$ do not contain other vector closed to the origin. So, at least in the neigbourhood of the origin, your result is true. However you might have other points. For instance if $N= S^1$ is a circle $exp_p^—1(p)$ conations infinitely many points.