12 problems

Let $(G,g)$ be a semi-Riemannian Lie group with a subgroup $K$ generating a left-invariant conformal foliation $\mathcal F$ on $G$ of codimension two. Semisimple subgroup minimalit…

Let $B\times_f^{N}F$ be an $N$-warped product of a Riemannian manifold $B$, a positive warping function $f$, and a weighted Riemannian manifold $F$. The weighted semi-Riemannian an…

Let $(M,g)$ be a 3-dimensional metric Lie group. A metric is complete when its geodesics are defined for all parameter values, and it has constant sectional curvature when every se…

Model-family conjecture. All $\mathcal{KS}_n$ semi-Riemannian manifolds are contained in the family of Theorem $$ . In particular, in the Lorentzian case, they coincide with their…

Parallel lightlike vector conjecture. If $(M,g)$ is a non-locally symmetric $\mathcal{KS}_n$ semi-Riemannian manifold, then it admits a parallel lightlike vector field.

Riemannian KS conjecture. There are no Riemannian $\mathcal{KS}_n$ manifolds of any order $r$, apart from the locally symmetric spaces.

Constant-curvature conjecture. If all Penrose limits of $(M,g)$ are flat, then $(M,g)$ has constant curvature.

Projective Lichnerowicz–Obata conjecture. Either $G$ acts on $M$ by affine transformations, or $(M^n,g)$ is Riemannian with positive constant sectional curvature.

Let $M$ be a $(2n+1)$-dimensional semi-Riemannian space form $M_{1}^{2n+1}(c)$ of index $1$, and let $B$ be a $2n$-dimensional Riemannian manifold. A proper biharmonic semi-Riemann…

Let $K_{s,\epsilon}$ be the semi-Riemannian quadric under consideration, and let $M\subset K_{s,\epsilon}$ be an algebraic hypersurface defined by a homogeneous polynomial $f$. Alg…

Let $M$ be a differentiable manifold and let $Z\subset TM$ be a countable set. Consider semi-Riemannian metrics $g$ on $M$ having a fixed index. For each $v\in Z$, let…

Periodic-geodesic genericity conjecture. Most of the genericity results proved in the paper should also hold for periodic geodesics, although the proof should use dynamical rather…