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DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES IN LORENTZ-MINKOWSKI SPACE

Year 2014, Volume: 7 Issue: 1, 44 - 107, 30.04.2014

Rafael López

https://doi.org/10.36890/iejg.594497
Cited By: 160

Abstract


Keywords

Lorentz-Minkowski space curve surface curvature Frenet mean curvature

References

  • [1] Abe, N., Koike, N., Yamaguchi, S., Congruence theorems for proper semi-Riemannian hyper- surfaces in a real space form, Yokohama Math. J. 35 (1987), 123–136.
  • [2] Barros, M., Caballero, M., Ortega, M., Rotational surfaces in L3 and solitons in the non-linear sigma model, Comm. Math. Phys. 290 (2009), 437–477.
  • [3] Bonnor, W. B., Null curves in a Minkowski space-time, Tensor (N. S.) 20 (1969), 229–242.
  • [4] Carmo, M. do, Differential Geometry of Curves and Surfaces, Prentice-Hall, Saddle River, 1976.
  • [5] Clelland, J. N., Totally quasi-umbilical timelike surfaces in R1,2, Asian J. Math. 16 (2012), no. 2, 189–208.
  • [6] Dillen, F., Kühnel, W., Ruled Weingarten surfaces in Minkowski 3-space, Manuscripta Math. 98 (1999), no. 3, 307-320.
  • [7] Ferrández, A., Gim´enez, A., Lucas, P., Null helices in Lorentzian space forms, Internat. J. Modern Phys. A 16 (2001), 4845–4863.
  • [8] Graves, L. K., Codimension one isometric immersions between Lorentz spaces, Trans. Amer. Math. Soc. 252 (1979), 367–392.
  • [9] Hano, J., Nomizu, K., Surfaces of revolution with constant mean curvature in Lorentz- Minkowski space, Tohoku Math. J. 36 (1984), 427–437.
  • [10] Inoguchi, J., Lee S., Null curves in Minkowski 3-space, International Elec. J. Geom. 1 (2008), 40–83.
  • [11] Klotz, T., Surfaces in Minkowski 3-space on which H and K are linearly related, Michigan Math. J. 30 (1983), 309–315.
  • [12] Kobayashi, O., Maximal surfaces in the 3-dimensional Minkowski space L3, Tokyo J. Math. 6 (1983), 297–309.
  • [13] Kühnel, W., Differential geometry. Curves – surfaces – manifolds. American Mathematical Society, Providence, RI, 2002.
  • [14] Liu, H., Translation surfaces with constant mean curvature in 3-dimensional spaces, J. Geom. 64 (1999), 141–149.
  • [15] López, R., Constant mean curvature surfaces with boundary in Euclidean three-space, Tsukuba J. Math. 23 (1999), 27–36.
  • [16] López, R., Constant mean curvature hypersurfaces foliated by spheres, Differential Geom. Appl., 11 (1999), 245–256.
  • [17] López, R., Cyclic hypersurfaces of constant curvature, Advanced Studies in Pure Mathemat- ics, 34, 2002, Minimal Surfaces, Geometric Analysis and Symplectic Geometry, 185–199.
  • [18] López, R., Differential Geometry of Curves and Surfaces in Lorentz-Minkowski Space, ArXiv:0810.3351 (2008).
  • [19] López, R., Constant Mean Curvature Surfaces with Boundary, Springer-Verlag, Berlin, 2013.
  • [20] López, R., Demir, E., Helicoidal surfaces in Minkowski space with constant mean curvature and constant Gauss curvature, to appear in Central Eur. J. Math.
  • [21] Magid, M., Lorentzian isoparametric hypersurface, Pacific. J. Math. 118, (1985), no. 1, 165– 197.
  • [22] Mira, P., Pastor, J. A., Helicoidal maximal surfaces in Lorentz-Minkowski space, Monatsh. Math. 140 (2003), 315–334.
  • [23] Montiel, S., Ros, A., Curves and Surfaces, Amer. Math. Soc., Graduate Studies in Math. 69. 2009.
  • [24] Nesovic, E., Petrovic-Torgasev, M., Verstraelen, L., Curves in Lorentzian spaces, Bolletino U.M.I. 8 (2005), 685–696.
  • [25] O’Neill, B., Elementary Differential Geometry, Academic Press, New York, 1966.
  • [26] O’Neill, B., Semi-Riemannian Geometry. With applications to relativity. Pure and Applied Mathematics, 103. Academic Press, Inc., New York, 1983.
  • [27] B. Riemann, Über die Flächen vom Kleinsten Inhalt be gegebener Begrenzung, Abh. Königl. Ges. Wiss. Gttingen, Math. Kl. 13 (1868), 329–333.
  • [28] Walrave, J., Curves and surfaces in Minkowski space, Thesis (Ph.D.), Katholieke Universiteit Leuven (Belgium), 1995.
  • [29] Weinstein, T., An Introduction to Lorentz Surfaces, de Gruyter Expositions in Mathematics, 22. Walter de Gruyter & Co., Berlin, 1996.
  • [30] Woestijne, I.W., Minimal surfaces in the 3-dimensional Minkowski space. Geometry and Topology of Submanifolds: II, Word Scientic Oress, 344–369, Singapore, 1990.

Year 2014, Volume: 7 Issue: 1, 44 - 107, 30.04.2014

Rafael López

https://doi.org/10.36890/iejg.594497
Cited By: 160

Abstract

References

  • [1] Abe, N., Koike, N., Yamaguchi, S., Congruence theorems for proper semi-Riemannian hyper- surfaces in a real space form, Yokohama Math. J. 35 (1987), 123–136.
  • [2] Barros, M., Caballero, M., Ortega, M., Rotational surfaces in L3 and solitons in the non-linear sigma model, Comm. Math. Phys. 290 (2009), 437–477.
  • [3] Bonnor, W. B., Null curves in a Minkowski space-time, Tensor (N. S.) 20 (1969), 229–242.
  • [4] Carmo, M. do, Differential Geometry of Curves and Surfaces, Prentice-Hall, Saddle River, 1976.
  • [5] Clelland, J. N., Totally quasi-umbilical timelike surfaces in R1,2, Asian J. Math. 16 (2012), no. 2, 189–208.
  • [6] Dillen, F., Kühnel, W., Ruled Weingarten surfaces in Minkowski 3-space, Manuscripta Math. 98 (1999), no. 3, 307-320.
  • [7] Ferrández, A., Gim´enez, A., Lucas, P., Null helices in Lorentzian space forms, Internat. J. Modern Phys. A 16 (2001), 4845–4863.
  • [8] Graves, L. K., Codimension one isometric immersions between Lorentz spaces, Trans. Amer. Math. Soc. 252 (1979), 367–392.
  • [9] Hano, J., Nomizu, K., Surfaces of revolution with constant mean curvature in Lorentz- Minkowski space, Tohoku Math. J. 36 (1984), 427–437.
  • [10] Inoguchi, J., Lee S., Null curves in Minkowski 3-space, International Elec. J. Geom. 1 (2008), 40–83.
  • [11] Klotz, T., Surfaces in Minkowski 3-space on which H and K are linearly related, Michigan Math. J. 30 (1983), 309–315.
  • [12] Kobayashi, O., Maximal surfaces in the 3-dimensional Minkowski space L3, Tokyo J. Math. 6 (1983), 297–309.
  • [13] Kühnel, W., Differential geometry. Curves – surfaces – manifolds. American Mathematical Society, Providence, RI, 2002.
  • [14] Liu, H., Translation surfaces with constant mean curvature in 3-dimensional spaces, J. Geom. 64 (1999), 141–149.
  • [15] López, R., Constant mean curvature surfaces with boundary in Euclidean three-space, Tsukuba J. Math. 23 (1999), 27–36.
  • [16] López, R., Constant mean curvature hypersurfaces foliated by spheres, Differential Geom. Appl., 11 (1999), 245–256.
  • [17] López, R., Cyclic hypersurfaces of constant curvature, Advanced Studies in Pure Mathemat- ics, 34, 2002, Minimal Surfaces, Geometric Analysis and Symplectic Geometry, 185–199.
  • [18] López, R., Differential Geometry of Curves and Surfaces in Lorentz-Minkowski Space, ArXiv:0810.3351 (2008).
  • [19] López, R., Constant Mean Curvature Surfaces with Boundary, Springer-Verlag, Berlin, 2013.
  • [20] López, R., Demir, E., Helicoidal surfaces in Minkowski space with constant mean curvature and constant Gauss curvature, to appear in Central Eur. J. Math.
  • [21] Magid, M., Lorentzian isoparametric hypersurface, Pacific. J. Math. 118, (1985), no. 1, 165– 197.
  • [22] Mira, P., Pastor, J. A., Helicoidal maximal surfaces in Lorentz-Minkowski space, Monatsh. Math. 140 (2003), 315–334.
  • [23] Montiel, S., Ros, A., Curves and Surfaces, Amer. Math. Soc., Graduate Studies in Math. 69. 2009.
  • [24] Nesovic, E., Petrovic-Torgasev, M., Verstraelen, L., Curves in Lorentzian spaces, Bolletino U.M.I. 8 (2005), 685–696.
  • [25] O’Neill, B., Elementary Differential Geometry, Academic Press, New York, 1966.
  • [26] O’Neill, B., Semi-Riemannian Geometry. With applications to relativity. Pure and Applied Mathematics, 103. Academic Press, Inc., New York, 1983.
  • [27] B. Riemann, Über die Flächen vom Kleinsten Inhalt be gegebener Begrenzung, Abh. Königl. Ges. Wiss. Gttingen, Math. Kl. 13 (1868), 329–333.
  • [28] Walrave, J., Curves and surfaces in Minkowski space, Thesis (Ph.D.), Katholieke Universiteit Leuven (Belgium), 1995.
  • [29] Weinstein, T., An Introduction to Lorentz Surfaces, de Gruyter Expositions in Mathematics, 22. Walter de Gruyter & Co., Berlin, 1996.
  • [30] Woestijne, I.W., Minimal surfaces in the 3-dimensional Minkowski space. Geometry and Topology of Submanifolds: II, Word Scientic Oress, 344–369, Singapore, 1990.
There are 30 citations in total.

Details

Primary Language English
Journal Section Research Article
Authors

Rafael López

Publication Date April 30, 2014
Published in Issue Year 2014 Volume: 7 Issue: 1

Cite

APA López, R. (2014). DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES IN LORENTZ-MINKOWSKI SPACE. International Electronic Journal of Geometry, 7(1), 44-107. https://doi.org/10.36890/iejg.594497
AMA López R. DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES IN LORENTZ-MINKOWSKI SPACE. Int. Electron. J. Geom. April 2014;7(1):44-107. doi:10.36890/iejg.594497
Chicago López, Rafael. “DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES IN LORENTZ-MINKOWSKI SPACE”. International Electronic Journal of Geometry 7, no. 1 (April 2014): 44-107. https://doi.org/10.36890/iejg.594497.
EndNote López R (April 1, 2014) DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES IN LORENTZ-MINKOWSKI SPACE. International Electronic Journal of Geometry 7 1 44–107.
IEEE R. López, “DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES IN LORENTZ-MINKOWSKI SPACE”, Int. Electron. J. Geom., vol. 7, no. 1, pp. 44–107, 2014, doi: 10.36890/iejg.594497.
ISNAD López, Rafael. “DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES IN LORENTZ-MINKOWSKI SPACE”. International Electronic Journal of Geometry 7/1 (April 2014), 44-107. https://doi.org/10.36890/iejg.594497.
JAMA López R. DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES IN LORENTZ-MINKOWSKI SPACE. Int. Electron. J. Geom. 2014;7:44–107.
MLA López, Rafael. “DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES IN LORENTZ-MINKOWSKI SPACE”. International Electronic Journal of Geometry, vol. 7, no. 1, 2014, pp. 44-107, doi:10.36890/iejg.594497.
Vancouver López R. DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES IN LORENTZ-MINKOWSKI SPACE. Int. Electron. J. Geom. 2014;7(1):44-107.

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https://doi.org/10.4134/JKMS.2015.52.3.523
Special equiform Smarandache curves in Minkowski space-time
Journal of the Egyptian Mathematical Society
https://doi.org/10.1016/j.joems.2017.04.003
A geometrical model of dishwasher spray arm for CornerWash
AIMS Mathematics
https://doi.org/10.3934/math.2022475
A different modelling of complex Hasimoto map for pseudo-null curves via Bishop frame
Complex Variables and Elliptic Equations
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Classification of zero mean curvature surfaces of separable type in Lorentz-Minkowski space
Tohoku Mathematical Journal
https://doi.org/10.2748/tmj.20210120a
On the pole indicatrix curve of the spacelike Salkowski curve with timelike principal normal in Lorentzian 3-space
Gümüşhane Üniversitesi Fen Bilimleri Enstitüsü Dergisi
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Surfaces of revolution with prescribed mean and skew curvatures in Lorentz-Minkowski space
Tohoku Mathematical Journal
https://doi.org/10.2748/tmj.20190729
Involutes of Pseudo-Null Curves in Lorentz–Minkowski 3-Space
Mathematics
https://doi.org/10.3390/math9111256
Moving frames and the characterization of curves that lie on a surface
Journal of Geometry
https://doi.org/10.1007/s00022-017-0398-7
Spacelike radial graphs of prescribed mean curvature in the Lorentz–Minkowski space
Analysis & PDE
https://doi.org/10.2140/apde.2019.12.1805
The geometry of Gauss map and shape operator in simply isotropic and pseudo-isotropic spaces
Journal of Geometry
https://doi.org/10.1007/s00022-019-0488-9
On the Regularity of the Minimizer of the Electrostatic Born–Infeld Energy
Archive for Rational Mechanics and Analysis
https://doi.org/10.1007/s00205-018-1331-4
On Characterizations of W-Directional Curves of Null Curves in Minkowski 4-Space
Süleyman Demirel Üniversitesi Fen Edebiyat Fakültesi Fen Dergisi
https://doi.org/10.29233/sdufeffd.902130
On spacelike equiform-Bishop Smarandache curves on S 1 2 $S_{1}^{2}$
Journal of the Egyptian Mathematical Society
https://doi.org/10.1186/s42787-019-0009-x
Timelike-Ruled and Developable Surfaces in Minkowski 3-Space E13
Frontiers in Physics
https://doi.org/10.3389/fphy.2022.838957
Timelike clad helices and developable surfaces in Minkowski 3-space
Rendiconti del Circolo Matematico di Palermo Series 2
https://doi.org/10.1007/s12215-018-0355-9
Characterization of spherical and plane curves using rotation minimizing frames
Boletim da Sociedade Paranaense de Matemática
https://doi.org/10.5269/bspm.49075
Elastic Sturmian spirals in the Lorentz-Minkowski plane
Open Mathematics
https://doi.org/10.1515/math-2016-0103
On geometric interpretations of split quaternions
Mathematical Methods in the Applied Sciences
https://doi.org/10.1002/mma.8518
Elementary Approachs on De Sitter Space
Mathematical Sciences and Applications E-Notes
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Some properties of parametric surfaces with a mixed-type curve
Optik
https://doi.org/10.1016/j.ijleo.2022.170240
Differential Equations of the Space-Like Loxodromes on the Helicoidal Surfaces in Minkowski 3-Space
Differential Equations and Dynamical Systems
https://doi.org/10.1007/s12591-016-0343-5
The two-dimensional analogue of the Lorentzian catenary and the Dirichlet problem
Pacific Journal of Mathematics
https://doi.org/10.2140/pjm.2020.305.693
Classification of f-Biharmonic Curves in Lorentz–Minkowski Space
Journal of Mathematics
https://doi.org/10.1155/2020/7529284
CONSTANT ANGLE RULED SURFACES DUE TO THE BISHOP FRAME IN MINKOWSKI 3-SPACE
Journal of Science and Arts
https://doi.org/10.46939/J.Sci.Arts-22.1-a09
Equiform spacelike normal curves according to equiform-Bishop frame in E13
Mathematical Methods in the Applied Sciences
https://doi.org/10.1002/mma.4618
Plane Lorentzian and Fuchsian Hedgehogs
Canadian Mathematical Bulletin
https://doi.org/10.4153/CMB-2014-053-7
On Constraint Manifolds of Lorentz Sphere
Analele Universitatii "Ovidius" Constanta - Seria Matematica
https://doi.org/10.2478/auom-2020-0017
Inextensible Flows of Null Cartan Curves in Minkowski Space R2,1
Universe
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Ruled and Rotational Surfaces Generated by Non-Null Curves with Zero Weighted Curvature in $(\mathbb{L}^{3},ax^{2}+by^{2})$
International Electronic Journal of Geometry
https://doi.org/10.36890/iejg.599817
Siacci's Theorem for Frenet Curves in Minkowski 3-Space
Mathematical Sciences and Applications E-Notes
https://doi.org/10.36753/mathenot.693053
THE SHAPE OPERATOR OF THE BEZIER SURFACES IN MINKOWSKI-3 SPACE
Journal of Science and Arts
https://doi.org/10.46939/J.Sci.Arts-20.4-a08
Harmonic evolutes of B-scrolls with constant mean curvature in Lorentz–Minkowski space
International Journal of Geometric Methods in Modern Physics
https://doi.org/10.1142/S0219887819500762
On an Indefinite Metric on a Four-Dimensional Riemannian Manifold
Axioms
https://doi.org/10.3390/axioms12050432
Minkowski-3 Uzayında Timelike Rasyonel Bezier Eğrilerinin Eğrilikleri Üzerine
Bitlis Eren Üniversitesi Fen Bilimleri Dergisi
https://doi.org/10.17798/bitlisfen.420286
ON SPHERICAL INDICATRICES AND THEIR SPHERICAL IMAGE OF NULL CURVES IN MINKOWSKI 3-SPACE
Journal of Universal Mathematics
https://doi.org/10.33773/jum.937457
Some Results on p-Shape Curvatures of Non-Lightlike Space Curves
International Electronic Journal of Geometry
https://doi.org/10.36890/iejg.545132
Evolutes of plane curves and null curves in Minkowski 3-space
Journal of Geometry
https://doi.org/10.1007/s00022-016-0334-2
The Lorentzian Version of a Theorem of Krust
Bulletin of the Malaysian Mathematical Sciences Society
https://doi.org/10.1007/s40840-020-00968-x
APPLICATIONS OF QUATERNIONS IN GALILEAN SPACES
Journal of Science and Arts
https://doi.org/10.46939/J.Sci.Arts-22.1-a12
A construction of timelike ruled surfaces in Minkowski three-space 𝔼13
Asian-European Journal of Mathematics
https://doi.org/10.1142/S1793557121501618
Elastic strips with null and pseudo-null directrix
International Journal of Geometric Methods in Modern Physics
https://doi.org/10.1142/S021988782050022X
One-Parameter Lorentzian Dual Spherical Movements and Invariants of the Axodes
Symmetry
https://doi.org/10.3390/sym14091930
Integrability aspects of the vortex filament equation for pseudo-null curves
International Journal of Geometric Methods in Modern Physics
https://doi.org/10.1142/S0219887817500906
Equilibrium measures and equilibrium potentials in the Born-Infeld model
Journal de Mathématiques Pures et Appliquées
https://doi.org/10.1016/j.matpur.2020.05.001
Accretive growth kinematics in Minkowski 3-space
International Journal of Geometric Methods in Modern Physics
https://doi.org/10.1142/S0219887817500694
The Geometry of the Inextensible Flows of Timelike Curves according to the Quasi-Frame in Minkowski Space R2,1
Symmetry
https://doi.org/10.3390/sym15030654
The Dirichlet Problem of the Constant Mean Curvature in Equation in Lorentz-Minkowski Space and in Euclidean Space
Mathematics
https://doi.org/10.3390/math7121211
Geometric Formulation and Multi‐dark Soliton Solution to the Defocusing Complex Short Pulse Equation
Studies in Applied Mathematics
https://doi.org/10.1111/sapm.12159
Nullcone Fronts of Spacelike Framed Curves in Minkowski 3-Space
Mathematics
https://doi.org/10.3390/math9222939
A zoo of translating solitons on a parallel light-like direction in Minkowski 3-space
Differential Geometry and its Applications
https://doi.org/10.1016/j.difgeo.2021.101796
https://doi.org/
On the Schrödinger map for regular helical polygons in the hyperbolic space
Nonlinearity
https://doi.org/10.1088/1361-6544/ac355c
On entire f-maximal graphs in the Lorentzian product Gn×R1
Journal of Geometry and Physics
https://doi.org/10.1016/j.geomphys.2016.12.023
Spacelike Lines with Special Trajectories and Invariant Axodes
Symmetry
https://doi.org/10.3390/sym15051087
A simple Gödel-type universe
International Journal of Geometric Methods in Modern Physics
https://doi.org/10.1142/S0219887823500184
Envelopes of circles and spacelike curves in the Lorentz–Minkowski 3-space
Forum Mathematicum
https://doi.org/10.1515/forum-2019-0092
Ricci-Curbastro Condition for Maximal Surfaces in the Lorentz-Minkowski Space
Results in Mathematics
https://doi.org/10.1007/s00025-016-0596-x
BERTRAND CURVES AND RAZZABONI SURFACES IN MINKOWSKI 3-SPACE
Bulletin of the Korean Mathematical Society
https://doi.org/10.4134/BKMS.2015.52.2.377
Accretive Darboux growth in Lorentz–Minkowski spacetime
Mathematical Methods in the Applied Sciences
https://doi.org/10.1002/mma.7227
On generalized null Mannheim curves in E24
Mathematical Methods in the Applied Sciences
https://doi.org/10.1002/mma.6375
A new approach to timelike surfaces which contain inclined curves as geodesics
Differential Equations and Dynamical Systems
https://doi.org/10.1007/s12591-019-00485-9
A Curvature Energy Problem on a Timelike Surface
International Electronic Journal of Geometry
https://doi.org/10.36890/iejg.545141
Null (Lightlike) $f$-Rectifying Curves in the Three Dimensional Minkowski Space $\mathbb{E}^3_1$
Fundamental Journal of Mathematics and Applications
https://doi.org/10.33401/fujma.708816
Ruled translating solitons in Minkowski 3-space
Journal of Geometry and Physics
https://doi.org/10.1016/j.geomphys.2021.104392
Elliptical rotations with hybrid numbers
Indian Journal of Pure and Applied Mathematics
https://doi.org/10.1007/s13226-022-00343-5
Dual transformations and quaternions
Mathematical Methods in the Applied Sciences
https://doi.org/10.1002/mma.7459