math.MG

We present an explicit piecewise linear map from a flat Klein bottle (i.e. one that is locally isometric to the Euclidean plane) into Euclidean 3-space an that is an isometric immersion -- a path isometry that is locally injective. The image is a self-intersecting polyhedron with embedded vertex figures where each vertex has zero angle defect. The construction of the map enforces the path isometry property so long as certain numerically-verifiable inequalities are satisfied, and we show that checking the local injectivity property at each vertex via another set of inequalities suffices. This work generalizes features from known piecewise linear isometric embeddings of flat tori and known piecewise smooth path isometries of flat Klein bottles, and apparently is the first explicit isometric immersion of a flat Klein bottle into $\mathbb{R}^3$.

We present an explicit piecewise linear map from a flat Klein bottle (i.e. one that is locally isometric to the Euclidean plane) into Euclidean 3-space an that is an isometric immersion -- a path isometry that is locally injective. The image is a self-intersecting polyhedron with embedded vertex figures where each vertex has zero angle defect. The construction of the map enforces the path isometry property so long as certain numerically-verifiable inequalities are satisfied, and we show that checking the local injectivity property at each vertex via another set of inequalities suffices. This work generalizes features from known piecewise linear isometric embeddings of flat tori and known piecewise smooth path isometries of flat Klein bottles, and apparently is the first explicit isometric immersion of a flat Klein bottle into $\mathbb{R}^3$.

We prove the coarea formula for Lipschitz maps from the subriemannian $n$th Heisenberg group $\mathbb H_n$ to $\mathbb R^{2n}$. Our result is new even when $n=1$ and provides the simplest vector-valued instance of the coarea formula in subriemannian geometry. This answers a question left open in the works of Magnani, Kozhevnikov, Magnani--Stepanov--Trevisan, and Julia--Nicolussi Golo--Vittone. The main difficulty of the proof is that a fiber of a $C^1_{\mathrm{H}}$ map $f: \mathbb H_n\to \mathbb R^{2n}$ is typically an unrectifiable curve. Its measure depends on the symplectic area of its projection to $\mathbb R^{2n}$. A bound on this area would imply the coarea formula, but examples of Kozhevnikov show that this area can be infinite or undefined. To overcome this, we introduce an integral that we use to define both the symplectic area of $\frac{1}{2}$--H\"older curves in $\mathbb R^{2n}$ and the symplectic area of projections of vertical curves in $\mathbb H_n$. Then, we give a geometric condition for this integral to converge. This yields, in addition, new results on the existence of the signed area of $\tfrac12$--H\"older planar curves that may be of independent interest. Finally, we use $\beta$--number estimates from the F\"assler--Orponen Dorronsoro Theorem to show that this geometric condition holds for almost every fiber.