Hirzebruch's 1954 Problem List — Problem 4

Definitions and mathematical statement

Identify Euclidean space $\mathbb{E}^4$ with ${\mathbb{R}}^4$ equipped with its standard inner product. Under the usual smooth embedded interpretation of a closed surface, let $M$ be a nonempty compact smooth two-dimensional manifold without boundary and let $f\colon M\hookrightarrow{\mathbb{R}}^4$ be a smooth embedding. Define the induced Riemannian metric by \[ g_p(v,w)=\langle {\,\mathrm{d}} f_p(v),{\,\mathrm{d}} f_p(w)\rangle, \qquad p\in M,\quad v,w\in T_pM, \] and write $K_g\colon M\to{\mathbb{R}}$ for its Gaussian curvature.

General existence question. Does there exist such a pair $(M,f)$ satisfying \[ \forall p\in M,\qquad K_g(p)<0? \] The source does not explicitly impose orientability on $M$ in this general question.

Genus-two question. In particular, does there exist such a pair $(M,f)$ with $M$ connected, orientable, and of genus $2$, satisfying the same strict curvature inequality at every point?

Interpretive qualifications. Here "closed" means compact and without boundary, and genus $2$ is interpreted in the usual orientable-surface sense. The supplied wording does not explicitly specify a differentiability class or distinguish embeddings from immersions. The displayed formulation uses smooth embeddings; whether an immersed formulation was also intended remains unresolved.

Short English statement

Can a closed surface embedded in Euclidean four-dimensional space have strictly negative Gaussian curvature everywhere for its induced metric? The source asks specifically whether this is possible for a surface of genus $2$. This draft uses the smooth embedded interpretation and requires human mathematical review.

Sources

• Local UnsolvedMath registry: permanent ID $800055$, exact source problem code catalogue entry, titled "Hirzebruch's 1954 Problem List --- Problem 4." The supplied record belongs to the AMR Open Problem Lists collection and labels the problem open; that label is supplied metadata, not an independently verified status assessment.
• Kotschick, Updates on Hirzebruch's 1954 Problem List (2013), Problem 4, https://arxiv.org/abs/1305.4623. This is the reference supplied by the record. The registry attributes the entry to Kotschick with proposed year 2013, while the title identifies the underlying list as Hirzebruch's 1954 list; these are distinct attribution contexts.
• Supplied provenance: extraction method arxiv_latex; source access date 2026-10-16. This access information is supplied curation and has not been independently verified for this draft.