Contractibility of Anosov Metric Spaces on Surfaces

Definitions and mathematical statement

The source leaves the topology on the metric space and the orientability convention implicit. The following is the conventional smooth formulation, conditional on interpreting the surface as connected and orientable and using the unquotiented space of smooth metrics with its $C^\infty$ topology. These conventions require confirmation against the source.

Let $M$ be a closed connected orientable smooth surface of genus $h\geq 2$. Here closed means compact without boundary. Let $\operatorname{Met}^\infty(M)$ denote the space of smooth Riemannian metrics on the fixed surface $M$, with the $C^\infty$ topology. For $g\in\operatorname{Met}^\infty(M)$, set \[ S_gM=\{(x,v)\in TM:g_x(v,v)=1\}, \] and let $\varphi_g^t:S_gM\to S_gM$ be its geodesic flow, with generating vector field $X_g$. This flow is Anosov if there is a continuous $D\varphi_g^t$-invariant splitting \[ T(S_gM)=E_g^s\oplus{\mathbb{R}} X_g\oplus E_g^u \] and constants $C\geq1$, $\lambda>0$ such that, for every $t\geq0$, \[ \|D\varphi_g^t(v)\|\leq Ce^{-\lambda t}\|v\|\quad(v\in E_g^s), \qquad \|D\varphi_g^{-t}(w)\|\leq Ce^{-\lambda t}\|w\|\quad(w\in E_g^u), \] where the norms come from an auxiliary Riemannian metric on $S_gM$.

Define the subspaces \[ \mathcal A(M)=\{g\in\operatorname{Met}^\infty(M):\varphi_g^t\text{ is Anosov}\}, \qquad \mathcal N(M)=\{g\in\operatorname{Met}^\infty(M):K_g(x)<0\text{ for every }x\in M\}, \] with their inherited topologies, where $K_g$ is Gaussian curvature. Negative curvature gives the inclusion $i:\mathcal N(M)\hookrightarrow\mathcal A(M)$.

Questions. For every such surface $M$:

1. Is $\mathcal A(M)$ contractible? Explicitly, do there exist $g_*\in\mathcal A(M)$ and a continuous map \[ H:\mathcal A(M)\times[0,1]\longrightarrow\mathcal A(M) \] satisfying $H(g,0)=g$ and $H(g,1)=g_*$ for every $g\in\mathcal A(M)$?
2. Is $i$ a weak homotopy equivalence? Explicitly, does it induce a bijection on path components and, for every $g\in\mathcal N(M)$ and every integer $n\geq1$, an isomorphism \[ i_*:\pi_n(\mathcal N(M),g)\longrightarrow\pi_n(\mathcal A(M),g)? \]

The second question occurs in the supplied original statement but is omitted from the supplied cleaned statement; it is retained here to preserve the original scope. The supplied background additionally states that the negative-curvature metrics deformation retract onto the hyperbolic metrics. That assertion is recorded as supplied background, without an independent verification or an inferred normalization of the word hyperbolic.

Short English statement

For each closed surface of genus at least two, ask whether the space of metrics with Anosov geodesic flow is contractible and whether including all negatively curved metrics preserves every homotopy group and the set of path components. This draft uses the conventional connected orientable smooth setting and the $C^\infty$ topology, subject to source confirmation. Both questions are retained from the original record.

Sources

• Local UnsolvedMath registry, supplied complete record for permanent ID $30003350$, source problem code catalogue entry, titled Contractibility of Anosov Metric Spaces on Surfaces, in the collection Oberwolfach Reports Open Problems. 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 literature-source label is "Oberwolfach report on Anosov metrics." The 2018 year in the citation and the 2017 year in the DOI and registry are preserved without reconciliation.
• Supplied literature source: "Farrell--Ontaneda on negatively curved metric spaces," DOI 10.4007/annals.2009.170.45. https://doi.org/10.4007/annals.2009.170.45. No more detailed bibliographic information is supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, states that negative-curvature metrics on closed surfaces deformation retract to hyperbolic metrics, and describes both contractibility of the Anosov-metric space and the weak homotopy-equivalence inclusion as unanswered. This is supplied dated curation, not an independently verified assessment.
• The registry assigns the statement status corrected_verified and reports that the official report passage was checked and rewritten while preserving its mathematical content. This is a supplied verification claim; no source visit or independent verification is asserted here.