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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: 173

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: 173

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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Geometric Properties in Minkowski Space-Time of Spacelike Smarandache Curves
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International Journal of Geometric Methods in Modern Physics
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Archiv der Mathematik
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Mathematical Methods in the Applied Sciences
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Screw Theory in Lorentzian Space
Advances in Applied Clifford Algebras
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Asymptotic Frame Fields of Rational Bézier Curves
Düzce Üniversitesi Bilim ve Teknoloji Dergisi
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Hyper-dual split quaternions and rigid body motion
Journal of Geometry and Physics
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TRANSLATION AND HOMOTHETICAL SURFACES IN EUCLIDEAN SPACE WITH CONSTANT CURVATURE
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Special equiform Smarandache curves in Minkowski space-time
Journal of the Egyptian Mathematical Society
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A geometrical model of dishwasher spray arm for CornerWash
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Complex Variables and Elliptic Equations
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Tohoku Mathematical Journal
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Gümüşhane Üniversitesi Fen Bilimleri Enstitüsü Dergisi
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Tohoku Mathematical Journal
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Involutes of Pseudo-Null Curves in Lorentz–Minkowski 3-Space
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Moving frames and the characterization of curves that lie on a surface
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Spacelike radial graphs of prescribed mean curvature in the Lorentz–Minkowski space
Analysis & PDE
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The geometry of Gauss map and shape operator in simply isotropic and pseudo-isotropic spaces
Journal of Geometry
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On the Regularity of the Minimizer of the Electrostatic Born–Infeld Energy
Archive for Rational Mechanics and Analysis
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Süleyman Demirel Üniversitesi Fen Edebiyat Fakültesi Fen Dergisi
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Journal of the Egyptian Mathematical Society
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Timelike-Ruled and Developable Surfaces in Minkowski 3-Space E13
Frontiers in Physics
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Timelike clad helices and developable surfaces in Minkowski 3-space
Rendiconti del Circolo Matematico di Palermo Series 2
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Boletim da Sociedade Paranaense de Matemática
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Elastic Sturmian spirals in the Lorentz-Minkowski plane
Open Mathematics
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Mathematical Methods in the Applied Sciences
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Elementary Approachs on De Sitter Space
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Some properties of parametric surfaces with a mixed-type curve
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Differential Equations of the Space-Like Loxodromes on the Helicoidal Surfaces in Minkowski 3-Space
Differential Equations and Dynamical Systems
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Pacific Journal of Mathematics
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Classification of f-Biharmonic Curves in Lorentz–Minkowski Space
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CONSTANT ANGLE RULED SURFACES DUE TO THE BISHOP FRAME IN MINKOWSKI 3-SPACE
Journal of Science and Arts
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Mathematical Methods in the Applied Sciences
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Plane Lorentzian and Fuchsian Hedgehogs
Canadian Mathematical Bulletin
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On Constraint Manifolds of Lorentz Sphere
Analele Universitatii "Ovidius" Constanta - Seria Matematica
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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
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Siacci's Theorem for Frenet Curves in Minkowski 3-Space
Mathematical Sciences and Applications E-Notes
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THE SHAPE OPERATOR OF THE BEZIER SURFACES IN MINKOWSKI-3 SPACE
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Harmonic evolutes of B-scrolls with constant mean curvature in Lorentz–Minkowski space
International Journal of Geometric Methods in Modern Physics
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On an Indefinite Metric on a Four-Dimensional Riemannian Manifold
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Bitlis Eren Üniversitesi Fen Bilimleri Dergisi
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ON SPHERICAL INDICATRICES AND THEIR SPHERICAL IMAGE OF NULL CURVES IN MINKOWSKI 3-SPACE
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Some Results on p-Shape Curvatures of Non-Lightlike Space Curves
International Electronic Journal of Geometry
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Evolutes of plane curves and null curves in Minkowski 3-space
Journal of Geometry
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The Lorentzian Version of a Theorem of Krust
Bulletin of the Malaysian Mathematical Sciences Society
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APPLICATIONS OF QUATERNIONS IN GALILEAN SPACES
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A construction of timelike ruled surfaces in Minkowski three-space 𝔼13
Asian-European Journal of Mathematics
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Elastic strips with null and pseudo-null directrix
International Journal of Geometric Methods in Modern Physics
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One-Parameter Lorentzian Dual Spherical Movements and Invariants of the Axodes
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Integrability aspects of the vortex filament equation for pseudo-null curves
International Journal of Geometric Methods in Modern Physics
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Equilibrium measures and equilibrium potentials in the Born-Infeld model
Journal de Mathématiques Pures et Appliquées
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Accretive growth kinematics in Minkowski 3-space
International Journal of Geometric Methods in Modern Physics
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The Geometry of the Inextensible Flows of Timelike Curves according to the Quasi-Frame in Minkowski Space R2,1
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The Dirichlet Problem of the Constant Mean Curvature in Equation in Lorentz-Minkowski Space and in Euclidean Space
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Geometric Formulation and Multi‐dark Soliton Solution to the Defocusing Complex Short Pulse Equation
Studies in Applied Mathematics
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Nullcone Fronts of Spacelike Framed Curves in Minkowski 3-Space
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A zoo of translating solitons on a parallel light-like direction in Minkowski 3-space
Differential Geometry and its Applications
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https://doi.org/
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On entire f-maximal graphs in the Lorentzian product Gn×R1
Journal of Geometry and Physics
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Spacelike Lines with Special Trajectories and Invariant Axodes
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A simple Gödel-type universe
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Envelopes of circles and spacelike curves in the Lorentz–Minkowski 3-space
Forum Mathematicum
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Ricci-Curbastro Condition for Maximal Surfaces in the Lorentz-Minkowski Space
Results in Mathematics
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BERTRAND CURVES AND RAZZABONI SURFACES IN MINKOWSKI 3-SPACE
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Accretive Darboux growth in Lorentz–Minkowski spacetime
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A new approach to timelike surfaces which contain inclined curves as geodesics
Differential Equations and Dynamical Systems
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A Curvature Energy Problem on a Timelike Surface
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Null (Lightlike) $f$-Rectifying Curves in the Three Dimensional Minkowski Space $\mathbb{E}^3_1$
Fundamental Journal of Mathematics and Applications
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Ruled translating solitons in Minkowski 3-space
Journal of Geometry and Physics
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Elliptical rotations with hybrid numbers
Indian Journal of Pure and Applied Mathematics
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