Research Article
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Year 2025, Volume: 18 Issue: 1, 33 - 47, 24.04.2025

Jun-ichi Inoguchi

https://doi.org/10.36890/iejg.1628997

Abstract

References

  • Benson, D. C.: An elementary solution of the brachistochrone problem, Amer. Math. Monthly 76 (8), 890-894 (1969). https://doi.org/10.2307/2317941
  • Besse, A. L.: Einstein Manifolds. Ergeb. Math. Grenzgeb. (3), 10, Springer-Verlag, 1987.
  • Bishop, R. L.: Clairaut submersions. in: Differential geometry (in honor of Kentaro Yano), Kinokuniya Book Store, Tokyo, 21-31 (1972).
  • Bishop, R. L.; O’Neill, B.: Manifolds of negative curvature. Trans. Amer. Math. Soc. 145, 1-49 (1969). https://doi.org/10.2307/1995057
  • Blair, D. E.: On a generalization of a catenoid. Canadian J. Math. 27, 231-236 (1975). https://doi.org/10.4153/CJM-1975-028-8
  • Brookfield, G.: Yet another elementary solution of the Brachistochrone problem. Math. Mag. 83:1, 59-63 (2010). https://doi.org/10.4169/002557010X480017
  • Chen, B-Y.: Geometry of Submanifolds and its Applications. Science University of Tokyo, Tokyo, 1981.
  • Chen, B-Y.: Geometry of warped products as Riemannian submanifolds and related problems. Soochow J. Math. 28 (2), 125-156 (2002).
  • Chen, B-Y.: On isometric minimal immersions from warped products into real space forms. Proc. Edinb. Math. Soc. (2) 45 (3), 579-587 (2002). https://doi.org/10.1017/S001309150100075X
  • Chen, B-Y.: Warped products in real space forms. Rocky Mountain J. Math. 34 (2), 551-563 (2004). https://doi.org/rmjm/1181069867
  • Chen, B-Y.: On warped product immersions. J. Geom. 82 (1-2), 36-49 (2005). https://doi.org/10.1007/s00022-005-1630-4
  • Chen, B-Y.: Geometry of warped product submanifolds: a survey. J. Adv. Math. Stud. 6 (2), 1-43 (2013).
  • Chen, B-Y.: Differential Geometry of Warped Product Manifolds and Submanifolds. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017.
  • Coll, V., Harrison, M.: Two generalizations of a property of the catenary. Amer. Math. Monthly 121 (2), 109–119 (2014). https://doi.org/10.4169/amer.math.monthly.121.02.109
  • Da Silva. L. C. B., Lopéz, R. I.: Catenaries in Riemannian surfaces. São Paulo J. Math. Sci. 18 (1), 389-406 (2024). https://doi.org/10.1007/s40863-023-00399-z Da Silva. L. C. B., Lopéz, R. I.: Catenaries and minimal surfaces of revolution in hyperbolic space. Proc. R. Soc. Edinb. A: Math., to appear. https://doi.org/10.1017/prm.2024.56
  • Delaunay, Ch.: Sur la surface de révolution dont la courbure moyenne est constante. J. Math. Pures Appl. 6, 309-314 (1841)
  • Dušek, Z., Kowalski, O.: Deformation properties of one remarkable hypersurface by H. Takagi in R4 . Math. Slovaca 58 (4), 483-496 (2008). https://doi.org/10.2478/s12175-008-0088-x
  • Ejiri, N.: A negative answer to a conjecture of conformal transformations of Riemannian manifolds. J. Math. Soc. Jpn. 33 (2), 261-266 (1981). https://doi.org/10.2969/jmsj/03320261
  • Ejiri, N.: A generalization of minimal cones. Trans. Amer. Math. Soc. 276, 347-360 (1983). https://doi.org/10.2307/1999438
  • Inoguchi, J.: Minimal surfaces in 3-dimensional solvable Lie groups. Chinese Ann. Math. B. 24, 73-84 (2003). https://doi.org/10.1142/S0252959903000086
  • Inoguchi, J.: Minimal surfaces in 3-dimensional solvable Lie groups. II. Bull. Austral. Math. Soc. 73, 365-374 (2006). https://doi.org/10.1017/S0004972700035401
  • Inoguchi, J.: Characteristic Jacobi operator on almost Kenmotsu 3-manifolds. Int. Electron. J. Geom. 16 (2), 464-525 (2023). https://doi.org/10.36890/iejg.1300339
  • Inoguchi, J., Lee, S.: A Weierstrass type representation for minimal surfaces in Sol. Proc. Amer. Math. Soc. 136 (6), 2209–2216 (2008). https://doi.org/10.1090/S0002-9939-08-09161-2
  • Kobayashi, S.: Differential Geometry of Curves and Surfaces, Springer Undergraduate Mathematics Series, 2019.
  • Kokubu, M.: On minimal submanifolds in product manifolds with a certain Riemannian metric, Tsukuba J. Math. 20 (1), 191-198 (1996). https://doi.org/10.21099/tkbjm/1496162989
  • Kruckovic, G. I.: On semireducible Riemannian spaces (in Russian). Dokl. Akad. Nauk SSSR (N.S.) 115, 862–865 (1957).
  • Kuczmarski, F., Kuczmarski, J.: Hanging around in non-uniform fields. Amer. Math. Monthly 122 (10), 941-957 (2015). https://doi.org/10.4169/amer.math.monthly.122.10.941
  • López, R.: The hanging chain problem in the sphere and in the hyperbolic plane. J. Nonlinear Sci. 34 (4), Paper No. 75, 23 p. (2024). https://doi.org/10.1007/s00332-024-10056-0
  • López, R.: A characterization of the catenary under the effect of surface tension. Rend. Circ. Mat. Palermo (2) 73 (3), 873-885 (2024). https://doi.org/10.1007/s12215-023-00956-7
  • López, R.: The hanging chain problem with respect to a circle. Kyungpook Math. J., to appear. arXiv:2405.12947v1[math.DG] (2024)
  • Matsuura, T., Matsuyama, T., Tanda, S.: Cycloid crystals by topology change. J. Crystal Growth 371, 17-22 (2013). https://doi.org/10.1016/j.jcrysgro.2013.01.043
  • Morita, M.: Brachistochrone curve as a geodesic in a surface (in Japanese), Bull. Nat. Int. Tech. Okinawa College 12, 37-45 (2018). https://www.okinawa-ct.ac.jp/old_site/UserFiles/File/04toshojoho_kakari/R1/kiyou/kiyou12.pdf
  • Nistor, A. I.: Constant angle surfaces in solvable Lie groups. Kyushu J. Math. 68 (2), 315-332 (2014). https://doi.org/10.2206/kyushujm.68.315
  • Nomizu, K.: On hypersurfaces satisfying a certain condition on the curvature tensor. Tôhoku Math. J. (2) 20, 46-59 (1968). https://doi.org/10.2748/tmj/1178243217
  • O’Neill, B.: Semi-Riemannian geometry with Applications to Relativity. Pure Appl. Math., 103 Academic Press, Inc., 1983.
  • Parker E.: A property characterizing the catenary, Math. Mag. 83 (1), 63-64 (2010). https://doi.org/10.4169/002557010X485120
  • Peñafiel, C., Quaglia, B. A., Trejos, H. A.: Elliptic Weingarten surfaces of minimal type in R2 ×h R. arXiv:2312.03527v1[math.DG] (2023)
  • Rojas, R.: The straight line, the catenary, the brachistochrone, the circle, and Fermat. arXiv:1401.2660v1 [math.HO] (2014)
  • Takagi, H.: An example of Riemannian manifolds satisfying R(X, Y ) · R = 0 but not ∇R = 0. Tôhoku Math. J. (2) 24, 105-108 (1972). https://doi.org/10.2748/tmj/1178241595

Catenaries, Cycloids and Warped Products

Year 2025, Volume: 18 Issue: 1, 33 - 47, 24.04.2025

Jun-ichi Inoguchi

https://doi.org/10.36890/iejg.1628997

Abstract

We study warped products derived from catenaries and cycloids. We give an example of non-homogeneous semi-symmetric 3-space closely related to cycloids.

Keywords

Warped product surface of revolution catenary catenoid cycloid

References

  • Benson, D. C.: An elementary solution of the brachistochrone problem, Amer. Math. Monthly 76 (8), 890-894 (1969). https://doi.org/10.2307/2317941
  • Besse, A. L.: Einstein Manifolds. Ergeb. Math. Grenzgeb. (3), 10, Springer-Verlag, 1987.
  • Bishop, R. L.: Clairaut submersions. in: Differential geometry (in honor of Kentaro Yano), Kinokuniya Book Store, Tokyo, 21-31 (1972).
  • Bishop, R. L.; O’Neill, B.: Manifolds of negative curvature. Trans. Amer. Math. Soc. 145, 1-49 (1969). https://doi.org/10.2307/1995057
  • Blair, D. E.: On a generalization of a catenoid. Canadian J. Math. 27, 231-236 (1975). https://doi.org/10.4153/CJM-1975-028-8
  • Brookfield, G.: Yet another elementary solution of the Brachistochrone problem. Math. Mag. 83:1, 59-63 (2010). https://doi.org/10.4169/002557010X480017
  • Chen, B-Y.: Geometry of Submanifolds and its Applications. Science University of Tokyo, Tokyo, 1981.
  • Chen, B-Y.: Geometry of warped products as Riemannian submanifolds and related problems. Soochow J. Math. 28 (2), 125-156 (2002).
  • Chen, B-Y.: On isometric minimal immersions from warped products into real space forms. Proc. Edinb. Math. Soc. (2) 45 (3), 579-587 (2002). https://doi.org/10.1017/S001309150100075X
  • Chen, B-Y.: Warped products in real space forms. Rocky Mountain J. Math. 34 (2), 551-563 (2004). https://doi.org/rmjm/1181069867
  • Chen, B-Y.: On warped product immersions. J. Geom. 82 (1-2), 36-49 (2005). https://doi.org/10.1007/s00022-005-1630-4
  • Chen, B-Y.: Geometry of warped product submanifolds: a survey. J. Adv. Math. Stud. 6 (2), 1-43 (2013).
  • Chen, B-Y.: Differential Geometry of Warped Product Manifolds and Submanifolds. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017.
  • Coll, V., Harrison, M.: Two generalizations of a property of the catenary. Amer. Math. Monthly 121 (2), 109–119 (2014). https://doi.org/10.4169/amer.math.monthly.121.02.109
  • Da Silva. L. C. B., Lopéz, R. I.: Catenaries in Riemannian surfaces. São Paulo J. Math. Sci. 18 (1), 389-406 (2024). https://doi.org/10.1007/s40863-023-00399-z Da Silva. L. C. B., Lopéz, R. I.: Catenaries and minimal surfaces of revolution in hyperbolic space. Proc. R. Soc. Edinb. A: Math., to appear. https://doi.org/10.1017/prm.2024.56
  • Delaunay, Ch.: Sur la surface de révolution dont la courbure moyenne est constante. J. Math. Pures Appl. 6, 309-314 (1841)
  • Dušek, Z., Kowalski, O.: Deformation properties of one remarkable hypersurface by H. Takagi in R4 . Math. Slovaca 58 (4), 483-496 (2008). https://doi.org/10.2478/s12175-008-0088-x
  • Ejiri, N.: A negative answer to a conjecture of conformal transformations of Riemannian manifolds. J. Math. Soc. Jpn. 33 (2), 261-266 (1981). https://doi.org/10.2969/jmsj/03320261
  • Ejiri, N.: A generalization of minimal cones. Trans. Amer. Math. Soc. 276, 347-360 (1983). https://doi.org/10.2307/1999438
  • Inoguchi, J.: Minimal surfaces in 3-dimensional solvable Lie groups. Chinese Ann. Math. B. 24, 73-84 (2003). https://doi.org/10.1142/S0252959903000086
  • Inoguchi, J.: Minimal surfaces in 3-dimensional solvable Lie groups. II. Bull. Austral. Math. Soc. 73, 365-374 (2006). https://doi.org/10.1017/S0004972700035401
  • Inoguchi, J.: Characteristic Jacobi operator on almost Kenmotsu 3-manifolds. Int. Electron. J. Geom. 16 (2), 464-525 (2023). https://doi.org/10.36890/iejg.1300339
  • Inoguchi, J., Lee, S.: A Weierstrass type representation for minimal surfaces in Sol. Proc. Amer. Math. Soc. 136 (6), 2209–2216 (2008). https://doi.org/10.1090/S0002-9939-08-09161-2
  • Kobayashi, S.: Differential Geometry of Curves and Surfaces, Springer Undergraduate Mathematics Series, 2019.
  • Kokubu, M.: On minimal submanifolds in product manifolds with a certain Riemannian metric, Tsukuba J. Math. 20 (1), 191-198 (1996). https://doi.org/10.21099/tkbjm/1496162989
  • Kruckovic, G. I.: On semireducible Riemannian spaces (in Russian). Dokl. Akad. Nauk SSSR (N.S.) 115, 862–865 (1957).
  • Kuczmarski, F., Kuczmarski, J.: Hanging around in non-uniform fields. Amer. Math. Monthly 122 (10), 941-957 (2015). https://doi.org/10.4169/amer.math.monthly.122.10.941
  • López, R.: The hanging chain problem in the sphere and in the hyperbolic plane. J. Nonlinear Sci. 34 (4), Paper No. 75, 23 p. (2024). https://doi.org/10.1007/s00332-024-10056-0
  • López, R.: A characterization of the catenary under the effect of surface tension. Rend. Circ. Mat. Palermo (2) 73 (3), 873-885 (2024). https://doi.org/10.1007/s12215-023-00956-7
  • López, R.: The hanging chain problem with respect to a circle. Kyungpook Math. J., to appear. arXiv:2405.12947v1[math.DG] (2024)
  • Matsuura, T., Matsuyama, T., Tanda, S.: Cycloid crystals by topology change. J. Crystal Growth 371, 17-22 (2013). https://doi.org/10.1016/j.jcrysgro.2013.01.043
  • Morita, M.: Brachistochrone curve as a geodesic in a surface (in Japanese), Bull. Nat. Int. Tech. Okinawa College 12, 37-45 (2018). https://www.okinawa-ct.ac.jp/old_site/UserFiles/File/04toshojoho_kakari/R1/kiyou/kiyou12.pdf
  • Nistor, A. I.: Constant angle surfaces in solvable Lie groups. Kyushu J. Math. 68 (2), 315-332 (2014). https://doi.org/10.2206/kyushujm.68.315
  • Nomizu, K.: On hypersurfaces satisfying a certain condition on the curvature tensor. Tôhoku Math. J. (2) 20, 46-59 (1968). https://doi.org/10.2748/tmj/1178243217
  • O’Neill, B.: Semi-Riemannian geometry with Applications to Relativity. Pure Appl. Math., 103 Academic Press, Inc., 1983.
  • Parker E.: A property characterizing the catenary, Math. Mag. 83 (1), 63-64 (2010). https://doi.org/10.4169/002557010X485120
  • Peñafiel, C., Quaglia, B. A., Trejos, H. A.: Elliptic Weingarten surfaces of minimal type in R2 ×h R. arXiv:2312.03527v1[math.DG] (2023)
  • Rojas, R.: The straight line, the catenary, the brachistochrone, the circle, and Fermat. arXiv:1401.2660v1 [math.HO] (2014)
  • Takagi, H.: An example of Riemannian manifolds satisfying R(X, Y ) · R = 0 but not ∇R = 0. Tôhoku Math. J. (2) 24, 105-108 (1972). https://doi.org/10.2748/tmj/1178241595
There are 39 citations in total.

Details

Primary Language English
Subjects Algebraic and Differential Geometry
Journal Section Research Article
Authors

Jun-ichi Inoguchi 0000-0002-6584-5739

Early Pub Date April 20, 2025
Publication Date April 24, 2025
Submission Date January 29, 2025
Acceptance Date April 1, 2025
Published in Issue Year 2025 Volume: 18 Issue: 1

Cite

APA Inoguchi, J.-i. (2025). Catenaries, Cycloids and Warped Products. International Electronic Journal of Geometry, 18(1), 33-47. https://doi.org/10.36890/iejg.1628997
AMA Inoguchi Ji. Catenaries, Cycloids and Warped Products. Int. Electron. J. Geom. April 2025;18(1):33-47. doi:10.36890/iejg.1628997
Chicago Inoguchi, Jun-ichi. “Catenaries, Cycloids and Warped Products”. International Electronic Journal of Geometry 18, no. 1 (April 2025): 33-47. https://doi.org/10.36890/iejg.1628997.
EndNote Inoguchi J-i (April 1, 2025) Catenaries, Cycloids and Warped Products. International Electronic Journal of Geometry 18 1 33–47.
IEEE J.-i. Inoguchi, “Catenaries, Cycloids and Warped Products”, Int. Electron. J. Geom., vol. 18, no. 1, pp. 33–47, 2025, doi: 10.36890/iejg.1628997.
ISNAD Inoguchi, Jun-ichi. “Catenaries, Cycloids and Warped Products”. International Electronic Journal of Geometry 18/1 (April 2025), 33-47. https://doi.org/10.36890/iejg.1628997.
JAMA Inoguchi J-i. Catenaries, Cycloids and Warped Products. Int. Electron. J. Geom. 2025;18:33–47.
MLA Inoguchi, Jun-ichi. “Catenaries, Cycloids and Warped Products”. International Electronic Journal of Geometry, vol. 18, no. 1, 2025, pp. 33-47, doi:10.36890/iejg.1628997.
Vancouver Inoguchi J-i. Catenaries, Cycloids and Warped Products. Int. Electron. J. Geom. 2025;18(1):33-47.
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