I'm curious about the following question and have not been able to find any literature on the topic: Suppose that $M$ is a closed negatively curved Riemannian manifold with a "large" quantity of totally geodesic submanifolds. Is there anything one can say about such a manifold?

To be more precise, consider such an $M$ (diffeomorphic to a complex hyperbolic manifold) carrying an almost complex structure $J$ which looks like a complex hyperbolic manifold in the following sense: for each $x \in M$ and each $v \in T_{x}M$ there is a totally geodesic surface $S$ tangent to the plane spanned by $v$ and $J(v)$ at $x$. It seems to be a natural question to ask if $M$ is itself complex hyperbolic (which would follow for example if $M$ was a Kahler manifold). I have no intuition as to whether this should be an easy or difficult problem so I'm curious (and thankful in advance) if anyone has some guidance/references.

Edit: I have made my question in the second paragraph more explicit below, since I think the original version may be ambiguous,

Let $X$ be a compact quotient of complex hyperbolic space $\mathbb{C} H^{n}$ ($n \geq 2$). Let $M$ be a Riemannian manifold obtained by taking a small smooth perturbation of the symmetric metric on $X$ (same underlying space, different metrics). Now suppose we are given an almost complex structure $J$ - not necessarily the standard one - such that for each $x \in M$ and each $v \in T_{x}M$ there is a totally geodesic subsurface $S$ of $M$ which is tangent to the plane spanned by $v$ and $J(v)$ inside of $T_{x}M$. Is $M$ isometric (up to rescaling) to $X$?

As far as I can tell from searching, Riemannian manifolds typically have very few higher dimensional totally geodesic submanifolds so the condition being imposed above is quite strong.

Edit 2: As pointed out in the comments, I actually want to assume these subsurfaces are also $J$-holomorphic, i.e., $TS \subset TM$ is preserved by $J$ for each surface $S$.

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