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Generalizations of the Theorems of Pappus-Guldin in the Heisenberg groups

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In this paper, we study areas (called p-areas) and volumes for parametric surfaces in the 3D-Heisenberg group \({\mathbb {H}}_1\), which is considered as a flat model of pseudo-hermitian manifolds. We derive the formulas of p-areas and volumes for parametric surfaces in \({\mathbb {H}}_1\) and show that the classical result of Pappus-Guldin theorems for surface areas and volumes hold if the surfaces satisfy some geometric properties. Some examples are also provided, including the surfaces with constant p-mean curvatures.

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References

  1. Allendoerfer, C.B.: The euler number of a riemann manifold. Am. J. Math. 62(1), 243–248 (1940)

    Article  MathSciNet  Google Scholar 

  2. Balogh, Zoltán M.: Size of characteristic sets and functions with prescribed gradient, Journal fur die Reine und Angewandte Mathematik, 63– 84, (2003)

  3. Bulmer-Thomas, I.: Guldin’s theorem-or pappus’s? Isis 75(2), 348–352 (1984)

    Article  MathSciNet  Google Scholar 

  4. Capogna, L., Danielli, D., Garofalo, N.: An isoperimetric inequality and the geometric sobolev embedding for vector fields. Math. Res. Lett. 1(2), 263–268 (1994)

    Article  MathSciNet  Google Scholar 

  5. Chang, S.-C., Chen, C.-W., Wu, C.-T.: On the cr analogue of reilly formula and yau eigenvalue conjecture. Asian J. Math. 22(5), 919–940 (2018)

    Article  MathSciNet  Google Scholar 

  6. Cheng, J.-H., Hwang, J.-F., Malchiodi, A., Yang, P.:, Minimal surfaces in pseudohermitian geometry, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze,, Minimal surfaces in pseudohermitian geometry, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze,, Minimal surfaces in pseudohermitian geometry, Annali della Scuola Normale Superiore di Pisa-Classe di ScienzeMinimal surfaces in pseudohermitian geometry. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze 4(1), 129–177 (2005)

    MathSciNet  Google Scholar 

  7. Cheng, J.-H., Hwang, J.-F., Yang, P.: Existence and uniqueness for p-area minimizers in the heisenberg group. Math. Ann. 337(2), 253–293 (2007)

    Article  MathSciNet  Google Scholar 

  8. Chiu, H.-L., Ho, P.T.: Global differential geometry of curves in three-dimensional heisenberg group and cr sphere. J. Geometr. Anal. 29(4), 3438–3469 (2019)

    Article  MathSciNet  Google Scholar 

  9. Chiu, H.-L., Huang, Y.-C., Lai, L.S.-H.: An application of the moving frame method to integral geometry in the heisenberg group, Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), 13, 097, pp27,2017,

  10. Chiu, H.-L., Lai, S.-H.: The fundamental theorem for hypersurfaces in heisenberg groups. Calc. Var. Partial. Differ. Equ. 54(1), 1091–1118 (2015)

    Article  MathSciNet  Google Scholar 

  11. Citti, G., Franceschiello, B., Sanguinetti, G., Sarti, A.: Sub-riemannian mean curvature flow for image processing. SIAM J. Imaging Sci. 9(1), 212–237 (2016)

    Article  MathSciNet  Google Scholar 

  12. Fenchel, W.: On total curvatures of riemannian manifolds: I. J. Lond. Math. Soc. 1(1), 15–22 (1940)

    Article  MathSciNet  Google Scholar 

  13. Flanders, H.: A further comment on pappus. Am. Math. Mon. 77(9), 965–968 (1970)

    Article  MathSciNet  Google Scholar 

  14. Franchi, B., Serapioni, R., Cassano, F.S.: Rectifiability and perimeter in the heisenberg group. Math. Ann. 321(3), 479–531 (2001)

    Article  MathSciNet  Google Scholar 

  15. Goodman, A.W., Goodman, G.: Generalizations of the theorems of pappus. Am. Math. Mon. 76(4), 355–366 (1969)

    Article  MathSciNet  Google Scholar 

  16. Gray, A., Miquel, V.: On pappus-type theorems on the volume in space forms. Ann. Glob. Anal. Geom. 18(3–4), 241–254 (2000)

    Article  MathSciNet  Google Scholar 

  17. Gual-Arnau, X., Vicente, M.: Pappus-guldin theorems for weighted motions. Bull. Belgian Math. Soc. Simon Stevin 13(1), 123–137 (2006)

    Article  MathSciNet  Google Scholar 

  18. Pursell, L.E.: More generalizations of a theorem of pappus. Am. Math. Mon. 77(9), 961–965 (1970)

    Article  MathSciNet  Google Scholar 

  19. Radelet-de Grave, P., Kepler, cavalieri, guldin. polemics with the departed, Seventeenth-century indivisibles revisited, Springer, 57– 86,2015,

  20. Ritoré, Manuel, Rosales, César, Rotationally invariant hypersurfaces with constant mean curvature in the heisenberg group \(h_n\), The Journal of Geometric Analysis, 16, 4, 703,2006,

  21. Ritoré, M., Rosales, C.: Area-stationary surfaces in the heisenberg group \(h_1\). Adv. Math. 219(2), 633–671 (2008)

    Article  MathSciNet  Google Scholar 

  22. Sanguinetti, G., Citti, G., Sarti, A.: Implementation of a model for perceptual completion in r 2\(\times \) s, Springer, International conference on computer vision and computer graphics, 188– 201,2008,

  23. Sharpe, R.W.: Differential geometry: Cartan’s generalization of klein’s erlangen program, Springer Science & Business Media, 166,2000,

  24. Vanhecke, L.: Geometry in normal and tubular neighborhoods, Rend. Sem. Fac. Sci. Univ. Cagliari, Supplement al, 58, 73– 176,1988,

  25. Weyl, H.: On the volume of tubes. Am. J. Math. 61(2), 461–472 (1939)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author would like to thank the anonymous reviewer for the comments for the regularity of the constructed surfaces. This work was funded in part by National Center for Theoretical Sciences (NCTS) in Taiwan and in part by Ministry of Science and Technology, Taiwan, with grant Number: 108-2115-M-024-007-MY2.

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  1. Department of Applied Mathematics, National University of Tainan, Tainan, Taiwan

    Yen-Chang Huang

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Huang, YC. Generalizations of the Theorems of Pappus-Guldin in the Heisenberg groups. J Geom Anal 31, 10374–10401 (2021). https://doi.org/10.1007/s12220-021-00649-6

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