Studying Scalar Curvature of Two Dimensional Kinematic Surfaces Obtained by Using Similarity Kinematic of a Deltoid

Affiliation(s)

^{1}
Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt.

^{2}
Department of Mathematics, Faculty of Science, Aswan University, Aswan, Egypt.

ABSTRACT

We consider a similarity kinematic of a deltoid by studying locally the scalar curvature for the corresponding two dimensional kinematic surfaces in the Euclidean space . We prove that there is no two dimensional kinematic surfaces with scalar curvature K is non-zero constant. We describe the equations that govern such the surfaces.

We consider a similarity kinematic of a deltoid by studying locally the scalar curvature for the corresponding two dimensional kinematic surfaces in the Euclidean space . We prove that there is no two dimensional kinematic surfaces with scalar curvature K is non-zero constant. We describe the equations that govern such the surfaces.

Cite this paper

Solouma, E. , Wageeda, M. , Gouda, Y. and Bary, M. (2015) Studying Scalar Curvature of Two Dimensional Kinematic Surfaces Obtained by Using Similarity Kinematic of a Deltoid.*Applied Mathematics*, **6**, 1353-1361. doi: 10.4236/am.2015.68128.

Solouma, E. , Wageeda, M. , Gouda, Y. and Bary, M. (2015) Studying Scalar Curvature of Two Dimensional Kinematic Surfaces Obtained by Using Similarity Kinematic of a Deltoid.

References

[1] Connor, O.J. and Robertson, E. (2006) Biography: Euler and Steiner.

http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Euler.html

[2] Barros, M. (1997) General Helices and a Theorem of Lancret. Proceedings of the American Mathematical Society, 125, 1503-1509.

http://dx.doi.org/10.1090/S0002-9939-97-03692-7

[3] Abdel-All, N.H. and Hamdoon, F.M. (2004) Cyclic Surfaces in E5 Generated by Equiform Motions. Journal of Geometry, 79, 1-11.

http://dx.doi.org/10.1007/s00022-003-1682-2

[4] Gordon, V.O. and Sement Sov, M.A. (1980) A Course in Descriptive Geometry. Mir Publishers, Moscow.

[5] Solouma, E.M. (2015) Three Dimensional Surfaces Foliated by an Equiform Motion of Pseudohyperbolic Surfaces in . JP Journal of Geometry and Topology, 0972-451x.

[6] Solouma, E.M. (2012) Local Study of Scalar Curvature of Two-Dimensional Surfaces Obtained by the Motion of Circle. Journal of Applied Mathematics and Computation, 219, 3385-3394.

http://dx.doi.org/10.1016/j.amc.2012.09.066

[7] Solouma, E.M., et al. (2007) Three Dimensional Surfaces Foliated by Two Dimensional Spheres. Journal of Egyptian Mathematical Society, 1, 101-110.

[8] Bottema, O. and Roth, B. (1990) Theoretical Kinematic. Dover Publications Inc., New York.

[9] Farin, G., Hoschek, J. and Kim, M. (2002) The Handbook of Computer Aided Geometric Design. North-Holland, Amsterdam.

[10] Castro, I. and Urbano, F. (1999) On a Minimal Lagrangian Submanifold of Cn Foliated by Spheres. Michigan Mathematical Journal, 46, 71-82.

http://dx.doi.org/10.1307/mmj/1030132359

[11] Jagy, W. (1998) Sphere Foliated Constant Mean Curvature Submanifolds. Rocky Mountain Journal of Mathematics, 28, 983-1015.

http://dx.doi.org/10.1216/rmjm/1181071750

[12] López, R. (2002) Cyclic Hypersurfaces of Constant Curvature. Advances Studies in Mathematics, 34, 185-199.

[13] Park, S.H. (2002) Sphere Foliated Minimal and Constant Mean Curvature Hypersurfaces in Space Forms Lorentz- Minkowski Space. Rocky Mountain Journal of Mathematics, 32, 1019-1044.

http://dx.doi.org/10.1216/rmjm/1034968429

[14] Kreyszig, E. (1975) Introduction to Differential Geometry and Riemannian Geometry. University of Toronto Press, Canada.

[15] Do Carmo, M. (1976) Differential Geometry of Curves and Surfaces. Prentice-Hall Inc., Englewood Cliffs.

[1] Connor, O.J. and Robertson, E. (2006) Biography: Euler and Steiner.

http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Euler.html

[2] Barros, M. (1997) General Helices and a Theorem of Lancret. Proceedings of the American Mathematical Society, 125, 1503-1509.

http://dx.doi.org/10.1090/S0002-9939-97-03692-7

[3] Abdel-All, N.H. and Hamdoon, F.M. (2004) Cyclic Surfaces in E5 Generated by Equiform Motions. Journal of Geometry, 79, 1-11.

http://dx.doi.org/10.1007/s00022-003-1682-2

[4] Gordon, V.O. and Sement Sov, M.A. (1980) A Course in Descriptive Geometry. Mir Publishers, Moscow.

[5] Solouma, E.M. (2015) Three Dimensional Surfaces Foliated by an Equiform Motion of Pseudohyperbolic Surfaces in . JP Journal of Geometry and Topology, 0972-451x.

[6] Solouma, E.M. (2012) Local Study of Scalar Curvature of Two-Dimensional Surfaces Obtained by the Motion of Circle. Journal of Applied Mathematics and Computation, 219, 3385-3394.

http://dx.doi.org/10.1016/j.amc.2012.09.066

[7] Solouma, E.M., et al. (2007) Three Dimensional Surfaces Foliated by Two Dimensional Spheres. Journal of Egyptian Mathematical Society, 1, 101-110.

[8] Bottema, O. and Roth, B. (1990) Theoretical Kinematic. Dover Publications Inc., New York.

[9] Farin, G., Hoschek, J. and Kim, M. (2002) The Handbook of Computer Aided Geometric Design. North-Holland, Amsterdam.

[10] Castro, I. and Urbano, F. (1999) On a Minimal Lagrangian Submanifold of Cn Foliated by Spheres. Michigan Mathematical Journal, 46, 71-82.

http://dx.doi.org/10.1307/mmj/1030132359

[11] Jagy, W. (1998) Sphere Foliated Constant Mean Curvature Submanifolds. Rocky Mountain Journal of Mathematics, 28, 983-1015.

http://dx.doi.org/10.1216/rmjm/1181071750

[12] López, R. (2002) Cyclic Hypersurfaces of Constant Curvature. Advances Studies in Mathematics, 34, 185-199.

[13] Park, S.H. (2002) Sphere Foliated Minimal and Constant Mean Curvature Hypersurfaces in Space Forms Lorentz- Minkowski Space. Rocky Mountain Journal of Mathematics, 32, 1019-1044.

http://dx.doi.org/10.1216/rmjm/1034968429

[14] Kreyszig, E. (1975) Introduction to Differential Geometry and Riemannian Geometry. University of Toronto Press, Canada.

[15] Do Carmo, M. (1976) Differential Geometry of Curves and Surfaces. Prentice-Hall Inc., Englewood Cliffs.