Open Problems and Questions about Geodesics — Question 20

Definitions and mathematical statement

Let $(M,g)$ be a closed Riemannian manifold, meaning compact and without boundary, of dimension $n>2$, with sectional curvature $K_g(\sigma)<0$ for every tangent two-plane $\sigma$. Write \[ SM=\{v\in TM:g(v,v)=1\} \] for its unit tangent bundle and $\varphi_t:SM\to SM$ for its geodesic flow. Its Anosov splitting is \[ T(SM)=E^s\oplus{\mathbb{R}} X\oplus E^u, \] where $X$ generates the flow, the subbundles are invariant under $D\varphi_t$, and there exist constants $C\geq1$ and $\lambda>0$ such that, for every $t\geq0$, \[ \|D\varphi_t\xi\|\leq Ce^{-\lambda t}\|\xi\| \quad(\xi\in E^s),\qquad \|D\varphi_{-t}\eta\|\leq Ce^{-\lambda t}\|\eta\| \quad(\eta\in E^u). \] Here the norms may be taken from the Sasaki metric on $SM$. Saying that $E^s$ and $E^u$ are $C^1$ means that they are $C^1$ subbundles over all of $SM$, equivalently that their associated sections of the Grassmann bundle are $C^1$.

Existence question (Burns--Matveev, Question 20). Does there exist such a manifold $(M,g)$ for which both $E^s$ and $E^u$ are $C^1$, but $g$ is not $1/4$-pinched?

Unresolved pinching qualification. The supplied record does not define $1/4$-pinching, so the last condition cannot be given a unique numerical interpretation from this record alone. For a global curvature interpretation, define \[ \kappa_{\min}=\min_{\sigma}(-K_g(\sigma)),\qquad \kappa_{\max}=\max_{\sigma}(-K_g(\sigma)),\qquad r(g)=\frac{\kappa_{\min}}{\kappa_{\max}}. \] These extrema range over all tangent two-planes, and $0<\kappa_{\min}\leq\kappa_{\max}$. If $1/4$-pinching means $r(g)\geq1/4$, the requested failure is $r(g)<1/4$; if it means $r(g)>1/4$, the requested failure is $r(g)\leq1/4$. A pointwise pinching interpretation would instead compare curvature extrema separately at each base point. The supplied evidence does not select among these interpretations.

Supplied context. The record states that the splitting is at least $C^1$ in dimension two or under $1/4$-pinching, and that in higher dimensions it is always Hölder continuous but typically not $C^1$. It refers to the introduction of $[\mathrm{Hasselblatt1994}]$ for regularity results. These are supplied contextual assertions; the pinching convention and the meaning of "typically" are not specified here.

Short English statement

Burns and Matveev ask whether a closed negatively curved Riemannian manifold of dimension greater than two can have both stable and unstable geodesic-flow distributions of class $C^1$ while failing $1/4$-pinching. The supplied record leaves the pinching convention, including its equality boundary, unresolved. Its registry status is supplied as open; that status has not been independently verified.

Sources

• Local UnsolvedMath registry, permanent ID $800940$, exact source problem code catalogue entry, titled Open Problems and Questions about Geodesics --- Question 20. Supplied attribution: Burns, Matveev; proposed year: 2019; supplied status: open. The record belongs to the AMR Open Problem Lists collection.
• Burns and Matveev, Open Problems and Questions about Geodesics, Question 20; the supplied source list labels the work 2019. Supplied source URL: https://arxiv.org/abs/1308.5417. The provenance records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance details, not a claim of independent access or verification.
• The statement cites $[\mathrm{Hasselblatt1994}]$, specifically its introduction, for a list of regularity results. No full bibliographic entry or link for this reference is supplied.
• Supplied upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82. The registry record was created and updated on 2026-10-16. These dates do not establish a current literature assessment.