Let $S^n$ be the $n$-dimensional round sphere (i.e. with Riemannian metric of constant curvature +1). Is there any classification result of totally geodesic embedded submanifolds? Are they all round spheres? How about general case when the curvature only assumes to be positive?


edit: To make the second question more precise: Let $S^n$ be the standard sphere with Riemannian metric such that the sectional curvature $\ge 1$. Let $N\in M$ be a totally geodesic connected submanifold. (assume $N$ is not a point). What is the possible topology of $N$?



Totally geodesic submanifold of round sphere
Contrary to another comment/answer, totally-geodesic $1$-manifolds in $S^3$ are not trivial because they can be disconnected. Great circles do not have to intersect, as can be seen by taking generic planes through the origin in $\mathbb{R}^4$. 


Geodesics sometimes can't be isotoped to fibers of the same Hopf fibration. Great circle links were the subject of [G. Walsh's thesis](https://gwalsh01.pages.tufts.edu/dissertationfinal.pdf). One chapter classified great circle links up to $5$ components. There are $1,1,2,3,7$, respectively. 


In higher dimensions, there are disconnected geodesic submanifolds of $S^n$ of dimension up to $(n-1)/2.$ The only place where you can have interesting linking is with totally geodesic $d$-spheres in $S^{2d+1}$, and I think the link theory is always nontrivial in that dimension.


