11 problems

Borisenko's conjecture. The zero vector field is the unique vector field whose image $\xi(M^n)$ is a totally geodesic submanifold of $TM^n$; equivalently, the base manifold is the…

Let $M$ be a closed, real-analytic Riemannian manifold with negative sectional curvature, and suppose that $M$ contains infinitely many closed totally geodesic immersed hypersurfac…

A non-arithmetic hyperbolic manifold is a hyperbolic manifold not arising from an arithmetic lattice; a totally geodesic hyperplane is a totally geodesic codimension-one submanifol…

Let $(M,g)$ be a closed Riemannian manifold with negative sectional curvature, of dimension $n\geq 3$. An immersed totally geodesic submanifold $N$ in $M$ is maximal if it is not c…

Let $(M,g)$ be a closed Riemannian manifold with negative sectional curvature, of dimension $n\geq 3$. An immersed totally geodesic submanifold $N$ in $M$ is maximal if it is not c…

Large totally geodesic hypersurface rigidity conjecture. For every $D>0$ and $k\in\mathbb{N}$, there exists $A=A(D,k)$ such that if $N$ contains a totally geodesic hyperbolic hyper…

Index Conjecture. There is a reflective submanifold $\Sigma$ of $M$ whose codimension equals the index of $M$:

Let $M$ be an irreducible Riemannian symmetric space. Its index $i(M)$ is the minimal possible codimension of a nontrivial totally geodesic submanifold. The reflective index…

Berndt–Olmos conjecture.

Type-IV total geodesy conjecture. Then

Rank-2 type-I total geodesy conjecture. If