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Correction to: Intrinsic curvature of curves and surfaces and a Gauss–Bonnet theorem in the Heisenberg group

  • Zoltán M. Balogh
  • Jeremy T. TysonEmail author
  • Eugenio Vecchi
Correction

1 Correction to: Math. Z. (2017) 287:1–38  https://doi.org/10.1007/s00209-016-1815-6

In the publication [1] there is an unfortunate computational error, which however does not affect the correctness of the main results.

Let us recall some notation from the paper. By \(\gamma : [a,b] \rightarrow {\mathbb {R}}^3\) we denote a \(\mathcal {C}^2\) smooth parametrized regular curve \(t\rightarrow \gamma (t)= (\gamma _1(t), \gamma _2(t), \gamma _3(t))\). The action of the standard contact form \(\omega = dx_3 -\frac{1}{2}(x_1dx_2- x_2dx_1)\) on \(\gamma \) is denoted by
$$\begin{aligned} \omega (\dot{\gamma }) = \omega (\dot{\gamma })(t)= \dot{\gamma }_{3}(t) - \dfrac{1}{2}\left( \gamma _{1}(t)\dot{\gamma }_{2}(t)-\gamma _{2}(t)\dot{\gamma }_{1}(t)\right) . \end{aligned}$$
A point \(t_0 \in [a,b]\) is called horizontal if and only if \(\omega (\dot{\gamma })(t_0)=0\). The mistake in the paper arises due to a statement implicitly assumed in the proof of Lemma 3.4, that at any horizontal point we also have that \(\omega (\ddot{\gamma })(t_0)=0\), where
$$\begin{aligned} \omega (\ddot{\gamma }) = \omega (\ddot{\gamma })(t)= \ddot{\gamma }_{3}(t) - \dfrac{1}{2}\left( \gamma _{1}(t)\ddot{\gamma }_{2}(t)-\gamma _{2}(t)\ddot{\gamma }_{1}(t)\right) . \end{aligned}$$
This fact is in general not true. As a result, various statements in the paper, including the second formula in equation (1.1), equation (3.4), the second part of equation (3.10), and the second displayed equations in both Lemma 4.8 and Proposition 4.13, do not hold for all horizontal points.

However, noticing that \(\omega (\ddot{\gamma })= \frac{d}{dt}\omega (\dot{\gamma })\) we see that the assertion \(\omega (\ddot{\gamma })(t_0)=0\) is still true for horizontal points that arise as accumulation points of other horizontal points. Since the parameterizing interval is compact, there are at most a finite number of isolated horizontal points \(t_1,\ldots , t_N\) at which the quantity \(\omega (\ddot{\gamma })(t_i)\) may be nonzero, and hence all of the preceding formulas hold at all points of [ab] except for this finite number of isolated points.

The main result of the paper, Theorem 1.1, is not affected by these corrections since its proof is based on an approximation argument relying on the Lebesgue dominated convergence theorem. In the application of this theorem a set of countably many points can be ignored as a null set, and the proof works as indicated in Section 6 of the paper.

Notes

Acknowledgements

We are grateful to Derek Jung and Maxim Tryamkin for pointing out the error in the proof of Lemma 3.4.

References

  1. 1.
    Balogh, Z.M., Tyson, J.T., Vecchi, E.: Intrinsic curvature of curves and surfaces and a Gauss-Bonnet theorem in the Heisenberg group. Math. Z. 287, 1–38 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Zoltán M. Balogh
    • 1
  • Jeremy T. Tyson
    • 2
    Email author
  • Eugenio Vecchi
    • 3
  1. 1.Mathematisches InstitutUniversität BernBernSwitzerland
  2. 2.Department of MathematicsUniversity of IllinoisUrbanaUSA
  3. 3.Dipartimento di MatematicaUniversità di BolognaBolognaItaly

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