Homotopy Equivalence of Negative Curvature and Anosov Metric Spaces

Definitions and mathematical statement

The supplied question is: "Are the spaces of negatively curved metrics and Anosov metrics on a manifold homotopy equivalent?" The record does not specify the class of manifolds, topology on the metric spaces, or whether a particular comparison map is intended. The following gives a precise conditional interpretation under customary closed-manifold conventions; these conventions are additional hypotheses, not hypotheses explicitly supplied by the source.

Fix a closed smooth manifold $M$ of dimension $n\geq 2$, where closed means compact without boundary. Let $\operatorname{Met}^{\infty}(M)$ denote the space of smooth Riemannian metrics on $M$, equipped with the $C^{\infty}$ topology on smooth symmetric covariant $2$-tensor fields. Define \[ \mathcal{R}_{<0}(M)=\{g\in\operatorname{Met}^{\infty}(M):K_g(\sigma)<0\text{ for every }x\in M\text{ and every two-dimensional subspace }\sigma\subset T_xM\}, \] where $K_g(\sigma)$ is sectional curvature.

For $g\in\operatorname{Met}^{\infty}(M)$, let \[ S_gM=\{v\in TM:g(v,v)=1\} \] be the unit tangent bundle, and let $\varphi_t^g:S_gM\to S_gM$ be its geodesic flow, with generating vector field $X_g$. Say that $g$ is an Anosov metric if there is a continuous $D\varphi_t^g$-invariant splitting \[ T(S_gM)=E^s\oplus{\mathbb{R}} X_g\oplus E^u \] and constants $C\geq1$, $\lambda>0$ such that, for every $z\in S_gM$, every $t\geq0$, and all $v^s\in E_z^s$, $v^u\in E_z^u$, \[ \|D\varphi_t^g(v^s)\|\leq Ce^{-\lambda t}\|v^s\|, \qquad \|D\varphi_{-t}^g(v^u)\|\leq Ce^{-\lambda t}\|v^u\|. \] Here the norms come from an auxiliary Riemannian metric on $S_gM$. Define \[ \mathcal{R}_{\mathrm A}(M)=\{g\in\operatorname{Met}^{\infty}(M):g\text{ is an Anosov metric}\}. \] Give both metric spaces their subspace topologies.

Under these conventions, the candidate question is whether \[ \mathcal{R}_{<0}(M)\simeq\mathcal{R}_{\mathrm A}(M). \] Explicitly, does there exist a pair of continuous maps \[ f:\mathcal{R}_{<0}(M)\longrightarrow\mathcal{R}_{\mathrm A}(M),\qquad h:\mathcal{R}_{\mathrm A}(M)\longrightarrow\mathcal{R}_{<0}(M) \] with $h\circ f\simeq\operatorname{id}_{\mathcal{R}_{<0}(M)}$ and $f\circ h\simeq\operatorname{id}_{\mathcal{R}_{\mathrm A}(M)}$? A distinct possible intended question asks whether the natural inclusion $\mathcal{R}_{<0}(M)\hookrightarrow\mathcal{R}_{\mathrm A}(M)$ is a homotopy equivalence. The supplied wording does not distinguish these targets.

The source also leaves unresolved whether the question ranges over every manifold in some specified class, concerns a fixed manifold, or assumes that one or both metric spaces are nonempty. No quotient by diffeomorphisms or rescaling has been taken in the conditional interpretation above; whether any such quotient was intended requires source review. The supplied status is open; it is a curation label, not an independently verified mathematical conclusion.

Short English statement

The question asks whether the space of smooth metrics with strictly negative sectional curvature has the same homotopy type as the space of metrics whose geodesic flow is Anosov. A precise conditional formulation is given for a closed smooth manifold using the $C^{\infty}$ topology. The source leaves the manifold hypotheses, quantification, nonemptiness assumptions, and intended comparison map unspecified. Its supplied open status requires independent review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003349$, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record labels the statement exact; its verification note concerns internal readability and byte-for-byte retention, not independent verification against the report. No individual proposer is supplied; the proposed year is recorded as $2017$.
• Supplied source citation: Mini-Workshop: Spaces and Moduli Spaces of Riemannian Metrics (2018), Oberwolfach Reports, pp. 133--166, DOI 10.4171/owr/2017/3. https://doi.org/10.4171/owr/2017/3. The supplied background attributes the problem to workshop open-problem discussions documented in Oberwolfach Reports. The title's $2018$ date and the record's $2017$ proposed year are retained without resolving their relationship.
• Supplied literature source: "Oberwolfach report on Anosov metrics." https://doi.org/10.4171/owr/2017/3.
• Supplied literature source: "Farrell--Ontaneda on pinched negatively curved metrics." https://doi.org/10.4007/annals.2009.170.45.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that high-dimensional nonconnectedness results show that the two metric spaces can differ at the level of path components, while no theorem establishes or refutes the proposed homotopy equivalence in general. This is supplied dated curation derived from the supplied OWR curation, not independently verified here. Its time-sensitive status and cited claims require independent verification; the record supplies no precise theorem or hypotheses supporting the path-component assertion.