5 problems

Let $X$ be a compact negatively curved manifold, let $\Delta_g$ be its Laplace operator, and let $\{\psi_{\lambda_j}\}_{j=1}^\infty$ be an orthonormal basis of eigenfunctions. For…

Let $N$ be a Riemannian manifold with sectional curvature at most $\delta<0$, and let $\gamma$ be a null-homotopic regular closed curve in $N$ whose length is greater than…

Let $N$ be a Riemannian manifold with sectional curvature bounded above by $\delta<0$, and let $\gamma$ be a regular closed curve in $N$. Langer–Singer conjecture. If … then $N$ ca…

Let $M$ and $N$ be closed negatively curved Riemannian manifolds. Write $\operatorname{CAT}(\mathrm{Diff},\mathrm{PL},\mathrm{Top})$-isomorphic to mean diffeomorphic, PL-homeomorph…

A locally compact group is compact-by-Lie if it has a compact normal subgroup such that the quotient is a Lie group. Let $G$ be a compactly generated locally compact group, and let…