9 problems

Let $M$ be a two-dimensional complete manifold, and fix an origin $O$. Say that its curvature is greater than or equal to an SL-bifurcator when it satisfies the comparison conditio…

Let $m,n$ be positive integers, with $m\ge n$ by symmetry, and consider the Kolmogorov flow on the torus with stream function $\psi=-\cos(mx)\cos(ny)$. Preston's conjecture. Conjug…

Let $f(x,y)=-\cos(mx)\cos(ny)$, where $m$ and $n$ are positive integers with $m\geq n$. A flow has a conjugate point if a conjugate point occurs along its geodesic. Conjugate-point…

Ma n 's conjecture. Then $M$ has no conjugate points.

No-conjugate-points conjecture. For any $(x_0,\mu)\in M\times\mathbb{R}^{*}$, the conjugate locus is empty.

Let $\operatorname{WF}^{\operatorname{SO}}$ and $\operatorname{WF}^{\operatorname{SE(2)}}$ denote the sub-Riemannian wavefronts in $\operatorname{SO}$ and $\operatorname{SE(2)}$, r…

A Maxwell point is a point along a geodesic reached by distinct geodesics of equal length, while a conjugate point is a point where the exponential mapping becomes singular. The co…

Chavel's conjecture. If every conjugate point of $\operatorname{G}/\operatorname{K}$ is isotropic, then $\operatorname{G}/\operatorname{K}$ is isometric to a Riemannian symmetric s…

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…